Sometimes there were 'parties' quite similar to our modern understanding of 'organised political party' in 'classical democracy Athens'. At least that is what from quite early on historians used as an analytical framework as well as we do to describe the situation. But the caveats are in the exact meaning, and how long lived such groupings were.
The word "party" is used in multiple ways, one of those being just 'a collective of supporters' — either of a certain person, argument, project. Most of the time very temporary.
If Athens was at War with Sparta, then we see two parties in a war. If in Athens most were in favour of going to that war but a minority remained in opposition to that war, we see again two 'parties'. But these were not stable political parties like in the modern sense in any way. Allegiances could switch very rapidly.
However, we do see in some ancient descriptions that over time an at first less visible rift solidified into a demarcation line for two relatively stable groups in Athens opposed to each other: named 'Democrats' and 'Oligarchs'.
This was developing under Pericles — and designed by Thucydides (Melesiou 'son of Melesias', not the historian).
We see this described in Plutarch, when Thucydides for the first time organised his supporters into a visibly separated political group acting as a collective to counter the overwhelming support the popular Pericles enjoyed:
Now there had been from the beginning a sort of seam hidden beneath the surface of affairs, as in a piece of iron, which faintly indicated a divergence between the popular and the aristocratic programme; but the emulous ambition of these two men cut a deep gash in the state, and caused one section of it to be called the "Demos" or the People and the other the "Oligoi" or the Few.
— Plutarch, Lives, Pericles, 11
This dichotomising portrayal seems to match the word usage in Thucydides and our modern understanding.
Only problem here: the historian Thucydides does not mention this special event, and it seems that analysing the depiction of other struggles in Athenian politics in Thucydides even precludes such fundamental polarisation between relatively stable classes poured into a corset we might call party. In Plutarch's depiction they were.
The reason why Plutarch and the overwhelming masses of later historians view (Athenian) politics through this refractory prism may be found in Aristotelian analytics: he described in his Politics events like the above as exemplifying an ideal type of political development, which was not an historical reconstruction, but rather a reductive and almost fictionalised account. The dichotomising tendency may be evident by the very titles Plutarch chose for his Lives.
For examples of this analytical raster in Aristotle compare Aristot. Pol. 1291b, 1279b34, 1288a, 1290a, 1304a, 1308b.
This raster as a distorting element of historical elements can be shown conclusively for another event: when Peisistratos established his tyrannis, we learn from Herodotus that there were three major factions at play, Coast people, Hill people, Plains people — which Aristotle describes as the 'typical way' (Aristot Pol. 1305b) this kind of constitution develops, since he describes for example the Coast people in his Athenaion politeia (Constitution of the Athenians) as politically abstractly motivated:
 The factions were three: one was the party of the Men of the Coast, whose head was Megacles the son of Alcmaeon, and they were thought chiefly to aim at the middle form of constitution; another was the party of the Men of the Plain, who desired the oligarchy, and their leader was Lycurgus; third was the party of the Hillmen, which had appointed Peisistratus over it, as he was thought to be an extreme advocate of the people.
— Aristot. Ath Pol. 13,4
Yet, this later account reads probably too much into the past, confirming to the framework of analysis, since in Herodotus we see just that: local groups supporting their leading aristocrat without any hint of constitutional abstracts:
 Hippocrates refused to follow the advice of Chilon; and afterward there was born to him this Pisistratus, who, when there was a feud between the Athenians of the coast under Megacles son of Alcmeon and the Athenians of the plain under Lycurgus son of Aristolaides, raised up a third faction, as he coveted the sovereign power. He collected partisans and pretended to champion the uplanders, and the following was his plan.
— Herodotus 1,5,3
If we now go back to Thucydides the historian and his word usage for 'parties' in a content manner that approaches a modern sense of peaceful struggle, we see a very concrete and not abstract usage that excludes stable frontlines and polarisations, characteristic for our modern party lines, but a very dynamic presentation. This is indeed not reasonably generalised when he speaks of 'those that are holding office', 'those that cannot hold office', of demos, oilgoi, plethos, dynathoi.
When Thucydides speaks of 'parties' in the word sense, then it is in an un-peaceful way of dealing with things, war or civil war:
Later on, one may say, the whole Hellenic world was convulsed; struggles being everywhere made by the popular chiefs to bring in the Athenians, and by the oligarchs to introduce the Lacedaemonians. In peace there would have been neither the pretext nor the wish to make such an invitation; but in war, with an alliance always at the command of either faction for the hurt of their adversaries and their own corresponding advantage, opportunities for bringing in the foreigner were never wanting to the revolutionary parties.
— Thuk. 3,82
Comparing different translations shows that "parties" itself is difficult to grasp in the Greek original, and variously translated rather freely as also factions etc.
For example 3,82,6 is:
6 καὶ μὴν καὶ τὸ ξυγγενὲς τοῦ ἑταιρικοῦ ἀλλοτριώτερον ἐγένετο διὰ τὸ ἑτοιμότερον εἶναι ἀπροφασίστως τολμᾶν: οὐ γὰρ μετὰ τῶν κειμένων νόμων ὠφελίας αἱ τοιαῦται ξύνοδοι, ἀλλὰ παρὰ τοὺς καθεστῶτας πλεονεξίᾳ. καὶ τὰς ἐς σφᾶς αὐτοὺς πίστεις οὐ τῷ θείῳ νόμῳ μᾶλλον ἐκρατύνοντο ἢ τῷ κοινῇ τι παρανομῆσαι.
and translated as:
 until even blood became a weaker tie than party, from the superior readiness of those united by the latter to dare everything without reserve; for such associations had not in view the blessings derivable from established institutions but were formed by ambition for their overthrow; and the confidence of their members in each other rested less on any religious sanction than upon complicity in crime.
But ἑταιρικός is:
A.of or befitting a companion : “ἡ ἑταιρική” companionship, Arist.EN1157b23 ; in full, ἑ. φιλία ib.1161b12. Adv. “-κῶς, προσφέρεσθαι” Id.EE1243a5.
2. τὸ ἑταιρικόν,=“ἑταιρεία” 1.2, Th.8.48 ; “ἑ. συνάγειν” Hyp.Eux.8 ; “τὰ ἑταιρικά” factions, clubs, Plu.Lys.5, D.C.37.57 ; = Lat. collegia, Id.38.13.
b. ties of party, opp. τὸ ξυγγενές, Th.3.82.
In other words: a much more temporary and ad hoc constellation than what we would consider a political party — and even in contrast to previous, more stable, allegiances, signifying the general breakdown for the political systems this war was in Thucydides' view.
For the general functions of the Athenian state in its fully developed form such a constellation of mortal enemies over trenches the general construction disincentivised such an emergent conflict: the basis was the ekklesia the general assembly, most important offices chosen by drawing lots, anually rotating and held accountable. This structurally prevented the development of a true professionalised political oligarchy to a very large extent, at least enough so that the oligoi may have been formed under Thucydides Melisou to counter the powerful rhetor Pericles. For a short time and not very sucessful. The result of this endeavour was imminent: Thucydides — as a single person, not 'his' 'group' — was ostracised…
To put this most bluntly: no, while the Athenian Democracy, radical and participatory, was functioning 'as intended' and working well, that is from the time of Solon to the catastrophic Sicilian Expedition, there was zero tolerance in the demos for any other 'party', as everyone speaking up usually was of an aristocratic background, yet had to always placate the demos and at least claim to be a philodemos, even later historians placed some individuals into a supposed oligoi 'camp', even though they still acted as individuals and in accordance to the whole demos. In this sense a post-Aristotelian view like that in Plutarch is misleading. Instigating a longer lived group, faction or 'party' was not possible as long as the system worked, which itself might be seen as conceived to prevent just that and of course another tyrannis.
That system ceased to work with the Peloponnesian War.
Cf the above to — Karl-Joachim Hölkeskamp: "Parteiungen und politische Willensbildung im demokratischen Athen: Perikles und Thukydides, Sohn des Melesias", Historische Zeitschrift, Vol 267, 1998, doi
Modem party-systems are founded on the premise that the operation of formal political groups does not go against the common interest. On the contrary, it is thought that parties are essential to free government. As is widely accepted, the Athenians did not entertain this pluralistic view.
Hence, the understanding that they held the operation of organised and impermeable political groups to be incompatible with democracy would account for the absence of relevant allusions in Aristotle's Politics. It would also explain the Athenians' general distaste for the activities of the hetaireiai and—to recall Pearson and his description of "party" rivalry as a mild form of it—for stasis.
Nevertheless, in certain instances, the Athenians had tried to transcend their own premises of democracy. The emergence of entities somewhat resembling political "parties" may be considered a centrifugal tendency, among others that appeared in their society. Insofar as it opposes the collectivity which absorbs—as Sartori puts it (1987.286-89)—individual political freedom, this tendency may be easily misinterpreted—and indeed it was—as an unexpectedly "modern" turn. However, the dilemma of whether or not "parties" did actually exist in the ancient period may be released from the narrow confines of a one-word answer. Instead, it may be more profitably discussed in terms of political mentalities, in both the ancient and modern contexts—conceptualised as unstable and often resembling something like a contradictory polyptych.
— V. I. Anastasiadis: "Political 'Parties' In Athenian Democracy: A Modernising 'Topos'", Arethusa, Fall 1999, Vol. 32, No. 3 (Fall 1999), pp.313–335. jstor