As currently phrased, the answer might be formulated as 'yes', sometimes they did, although probably not 'in general'. Pater was not used like a synonym for 'priest' like it is now often done in Christian faith communications. Neither concepts nor word usage are entirely congruent or compatible, despite Wiktionary listing this (possible) translation:
pater m (genitive patris); third declension
- honorific title
Lewis&Short for example do not list this example…
The 'yes' would apply to for example certain fetials:
The duties of the fetials included advising the Senate on foreign affairs and international treaties, making formal proclamations of peace and of war, and confirming treaties. They also carried out the functions of traveling heralds or ambassadors (Pater Patratus).
Which is strictly speaking a title and not a form of address.
At another Imperial-era, Lavinium-based festival, the pater patratus of Lavinium made sacrifices to celebrate the 338 BCE treaty between Rome and Lavinium.
— Claudia Moser: "The Altars of Republican Rome and Latium. Sacrifice and the Materiality of Roman Religion", Cambridge University Press: Cambridge, New York, 2018, p57.
The envoys, the fetiales, were sacralized, looked upon as pater patratus and “priests,” […]
— Jörg Rüpke: "Pantheon. A New History of Roman Religion", Princeton University Press: Princeton & Oxford, 2018, p112.
In Mithraism, any leader of a Mithraeum was called pater. This would be the best match for usage in Christian circles.
Casting a wider net, for example both Caesar and Augustus were pontifex maximus and at the same time pater patriae.
Equally mixed up would have been then the most common usage of priestly duties for the 'office'/function of pater familias:
As priest of familia, gens and genius
The domestic responsibilities of the pater familias included his priestly duties (sacra familiae) to his "household gods" (the lares and penates) and the ancestral gods of his own gens. The latter were represented by the di parentes as ancestral shades of the departed, and by the genius cult. Genius has been interpreted as the essential, heritable spirit (or divine essence, or soul) and generative power that suffused the gens and each of its members. As the singular, lawful head of a family derived from a gens, the pater familias embodied and expressed its genius through his pious fulfillment of ancestral obligations. The pater familias was therefore owed a reciprocal duty of genius cult by his entire familia. He in his turn conferred genius and the duty of sacra familiae to his children—whether by blood or by adoption.
Roman religious law defined the religious rites of familia as sacra privata (funded by the familia rather than the state) and "unofficial" (not a rite of state office or magistracy, though the state pontifices and censor might intervene if the observation of sacra privata was lax or improper). The responsibility for funding and executing sacra privata therefore fell to the head of the household and no other. As well as observance of common rites and festivals (including those marked by domestic rites), each family had its own unique internal religious calendar—marking the formal acceptance of infant children, coming of age, marriages, deaths and burials. In rural estates, the entire familia would gather to offer sacrifice(s) to the gods for the protection and fertility of fields and livestock. All such festivals and offerings were presided over by the pater familias.
At least from hellenistic times onward Greek usage of words for family-relations was indeed rather widespread and common, even involving quite colloquial variations:
Civic bodies and other organizations commonly honored a benefactor or functionary by referring to him or her as "father (patres) of the polis", "mother" […]
There are even a number of cases where the more colloquial and affectionate term "papa" or "daddy" (παππας or αππας in Greek and variants) is used of religious leaders in associations devoted to the mysteries and in other groups. In some cases when members of an association regularly referred to their leader as "mother," "father," or even "papa," I suggest that they were alluding to the same sort of family atmosphere within the group that the term "brothers" or "sisters" would evoke.
— Philip A. Harland: "Familial Dimensions of Group Identity: "Brothers" (ΑΔΕΛΦΟΙ) in Associations of the Greek East", Journal of Biblical Literature, Vol. 124, No. 3, , 2005), pp. 491–513. jstor
For a more general approach:
Consider, for instance, how appellations like 'father', 'saviour', etc., might have been understood. There was nothing legal or definitive about them. […]
Our evidence shows that the father analogy was used
regularly of Greek founders and saviours, at least in literary if not epigraphic evidence. It is clear that the image of the state as a family does not derive from normal social institutions, for an examination of these leads to the opposite conception of the state as a collection of families (III). Finally, employment of the father analogy in Greek kingship literature suggests the applicability of the general model and provides background to Roman usage (IV).
— T. R. Stevenson: "The Ideal Benefactor and the Father Analogy in Greek and Roman Thought", Classical Quarterly, Vol 42, No 2, 1992, pp 421–436. doi
In Roman-style collegia some officials were called pater — and mater — with also fratres and other titles from this kind of kinship vocabulary:
Other associations appear to have had a more differentiated leadership, with magistri or quinquennales as the highest positions, then other officials variously named mater collegii, pater collegii, curator, honoratus, immunis, quaestor, sacerdos and scriba. […]
Although a typical Roman or Ostian Mithraist is more difficult to characterize, members do frequently seem to belong to those organizational structures of public service and imperial or private familiae which were the fabric of the empire. Dangerous though it is to do so, one might take as the quintessential city mithraeum that in —and of— the police barracks for seconded detachments (the Castra Peregrinorum), and as the quintessential city Mithraist the imperial freedman and father and priest of the unconquered Mithras of the imperial household (domus Augustanae) L. Septimius Archelaus,5or the dedicant of the earliest known icon of the bull-killing Mithras, Alcimus, the slave vilicus (bailiff) of Trajan’s Praetorian Prefect Ti. Claudius Livianus.
— John S. Kloppenborg & Stephen G. Wilson (eds): "Voluntary Associations in the Graeco-Roman World", Routledge: London, New York, 1996.
Since the question seems to aim at a comparison with Christian word usage for father, it may also be of interest that also Jews were frequently called 'father [of the synagogue]':
(Cl] 694): “[Claudius] Tiberius Polycharmus also called Achyrius, father of the synagogue in Stobi … built the building for the holy place and the dining room, together with the colonnaded hall, from his own funds, without touching in any way the funds of the sanctuary. But the complete right to and ownership of the upper story I … reserve for myself …”
Similarly, addressing any god or deity featured frequently the form of address as 'father':
The study does not include nouns or adjectives occurring only in epithets, such as pater, which, despite implicit religious concepts, function primarily as respectful terms of address. […]
Cato's earlier prayer for the lustration of a farm shows use of the formula in a private setting: Mars pater, te precor quaesoque (Rust. 141).
— Frances V. Hickson: "Roman Prayer Language Livy and the Aneid of Vergil", –Beiträge zur Altertumskunde 30,_ Springer: Wiesbaden, 1993.
Also compare the very etymology of Jupiter:
Old Latin from Proto-Italic *djous patēr (“Jupiter”, literally “Sky Father”)
The Christian practice is therefore not only inspired by Jewish heritage, just like the parallel Greek usage of elder, presbyter:
Exodus 18:12: και έλαβεν Ιοθώρ ο γαμβρός Μωυσή ολοκαυτώματα και θυσίας τω θεώ παρεγένετο δε Ααρών και πάντες οι πρεσβύτεροι Ισραήλ συμφαγείν άρτον μετά του γαμβρού Μωυσή εναντίον του θεού
but around the mediterranean world for concepts of father as well:
Furthermore, parental language, such as “mother of the synagogue” or “father of the association,” was another important way in which members within Judean gatherings and other associations expressed social hierarchies and identified with other members of the group (ch. 4). The Judean use of such parental terminology mirrors similar practices within Greek cities in the Roman Empire, pointing to one instance of acculturation to the practices of civic communities generally and associations specifically. […]
Within the realm of honours in the Greek East and Asia Minor in particular, it was not unusual for civic bodies and other organizations to express honour for, or positive relations with, a benefactor or functionary by referring to him or her as “father” (πατήρ) “mother” (μήτηρ), “son” (υἱός), “daughter” (θυγάτηρ), “foster-father” (τροφεύς), or “foster-child” (τρόφιμος). Evidence for this usage begins as early as the second century BCE (as at Teos involving “fathers”) […]
There are several examples of such paternal or maternal terminology from Greece, sometimes in reference to important religious functionaries. In the Piraeus (port city to Athens) there was an organization in honour of Syrian deities and the Great Mother whose leadership included a priest, a priestess, a “horse,” and a “father of the orgeonic synod” (SIG3 1111 = IG III 1280a, esp. line 15; ca. 200–211 CE). The “father” is listed alongside these other functional roles without any suggestion that this is merely an honorific title. In connection with Syria, it is worth mentioning the “father of the association” (κοίνου) that set up a monument near Berytos (IGR III 1080). […]
Another later instance from this region involves a “company” (σπεῖρα) of Dionysos worshipers in nearby Histria. Here the group is also designated as “those gathered around” (οἱ περὶ) their “father,” Achilleus son of Achillas, their priest, and their hierophant in a way that suggests that all three were also members with functional roles within the group (IGLSkythia I 99; 218–222 CE). The same man was also the “father” of what seems to be a different group called the “hymn-singing elders (ὑμνῳδοὶ πρεσβύτε|ροι) gathered around the great god Dionysos” (IGLSkythia I 100, lines 4–5, 10–11). If this was not enough, he was also the “father” of a third association, this one devoted to the Great Mother at Tomis. There he is listed between a priest and a chief tree-bearer (ἀρχιδενδροφόρος), both figures with functional roles in cultic activities of the group (IGLSkythia II 83). […]
The use of parental language for benefactors and leaders is not limited to Dionysiac groups, then. A board of temple-wardens (νεωκόροι) devoted to Saviour Asklepios in Pautalia, Thracia (southwest of Serdica), refers to the leader of the group simply as “the father.” […]
One inscription from Rome involves a group founded by Greek-speaking immigrants from Alexandria devoted to Sarapis (IGUR 77 = SIRIS 384; 146 CE). This “sacred company (τάξις) of the Paianistai” devoted to “Zeus Helios, the great Sarapis, and the revered (σέβαστοι) gods” honours Embe, who is called both “prophet” and “father of the company.” The use of the term prophet here strongly suggests an active role for this “father” within the group. […]
The “sacred, athletic, wandering, world-wide association,” which was devoted to the god Herakles (see a photo of this god in figure 9), had a significantly long history. Originally based in Asia Minor (probably at Ephesos), the headquarters of this guild (which also had local branches at various locations in the East) was moved to Rome some time in the second century, probably around 143 CE. A Greek inscription from the time of Constantine reveals that, at least by this time and likely earlier as well, the members of this “world-wide” organization expressed positive connections with fellow-members using familial metaphors. Well-respected members are repeatedly called “our brother” (τοῦ ἀδελφοῦ ἡμῶν) in the inscription and the high priest of the guild is called “our father” (τοῦ πατρός ἡμῶν) (IGUR 246 = IG XIV 956B, lines 11, 12, 14).
— Philip A. Harland: "Dynamics of Identity in the World of the Early Christians. Associations, Judeans, and Cultural Minorities", t&t Clark: New York, London, 2009.