Pre-modern empires are generally driven toward territorial expansion. Obviously the territories of the Mediterranean and Central Asia each brought a range of different resources that the opposing empire would like to control. When two large rivals get in each others way, conflict is almost inevitable. It is tempting to draw comparisons with other bipolar situations in the ancient world, like Athens and Sparta. If one were significantly weaker then the other, the rival would have been erased much sooner. But when two large powers emerge, a stable situation of lower-level conflict can emerge and will continue until one gets the upper hand.
However, as again is more generally the case, the two civilizations were not in a constant state of all-out war. In that sense, they did achieve a high degree of peaceful co-existence at various moments in time. An interesting illustration of that is the degree of cultural mixing that occurred in the borderzone between the two empires. Quoting an article from The Metropolitan Museum of Art:
The Sasanian empire expanded its geographic scope dramatically under
Shapur I (r. 241–72), when its lands stretched into Central Asia and
stopped just short of the Mediterranean in current-day Syria. This
began a centuries-long cycle of expansion and retraction, which placed
the Sasanians in direct conflict with Rome, and later, Byzantinum.
Much of the eastern Mediterranean became a buffer zone between
empires, which presented opportunities for rich cultural exchanges in
between devastating military battles. Trade and the movement of people resulted in a distinctive frontier culture in towns like Palmyra, Dura Europas, and Resafa that mixed Mediterranean-Roman and Hellenistic-Parthian elements. Dress preferences reflect these contacts. Sculptures from Palmyra, for instance, depict some men wearing pants associated with eastern dress styles, while others don the togas of Roman elites.