The scope of your question covers a massive swathe of history. Using the broadest range of dates, it covers a period from 247 B.C. to 651 C.E., which is just a touch under 900 years. Rather than try to provide an overview of this entire period, I will discuss the borderland regions during the end of this time period, as both the Sassanian and Byzantine empires began to wane, but prior to the emergence of Islam; lets say 500-630 C.E.. While we refer to this period as the Byzantine empire, it is worth noting that there was no firm transition date from Rome to Byzantium, and at this stage in history the Emperors of Constantinople (with fair justification) considered themselves to be a part of the Imperial lineage that extended back to Aeneas, Romulus, Lucretia, Lucius Brutus, and the other so-called Gods of their legends.

Rather than get right into the sources, I can't help but begin by quoting a passage from R. Scott Bakker's "The Darkness That Comes Before", a fantasy novel (Not a historical source!) that nevertheless perfectly summarizes the relationship between the two powers (Byzantine and Sassanid):

When two civilized peoples find themselves at war for centuries, any number of common interests will arise in the midst of their greater antagonism. Ancestral foes share many things: mutual respect, a common history, triumph in stalemate, and a plethora of unspoken truces.

Such was the situation with the Sassanian and Byzantine empires prior to their death/slow march into decrepitude. By this period years of fighting had led to a situation, that, while not quite a stalemate, as both powers still took every opportunity to push any advantage they stumbled upon, had the two empires coming to regard one another as eternal antagonists whose struggles, like the tides, waxed and waned but simultaneously remained eternal and unceasing.

To the East were the Persians [Sassanids], whom the Romans had not beaten in battle for generations. The Persians had no desire to destroy the Empire; they considered themselves and the Romans the "Twin Eyes of the World". The Romans held the barbarians in the west from the incursions on the Persian Empire (as the Persians guarded the east). Additionally, the Romans were a source of great trade and the leaders of both empires were on friendly terms. Both used each other as a source of intelligence and for assistance in internal court intrigues.

Major Anthony Brogna, "The Generalship of Belisarius"

Given that the border regions had not remained constant over the past 900 years of strife (if not moreso, many Romans considered themselves heir to the tradition of Alexander and his campaigns against the Persian empire proper, which in turn had its roots in the ancient Greco-Persian wars), it should not be surprising that there was, by the sixth century, a rich cross-cultural pollination in the regions commonly contested by the two powers. While the courts or both empires were staunchly Christian and Zoroastrian at this point, the ebb and flow of conquest led to deposits of the competing ideologies remaining behind even after their proponents retreated. As a general rule, the Sassanians, being more religiously tolerant during era I am discussing, found themselves ruling a much more cosmopolitan empire. This was not always the case, however. A perfect example of the Zoroastrian influence Rome absorbed as it battled the earlier Parthian empire, prior to the heavy-handed Christian evangelisation of Justinian and Theodosius II, was the tremendous popularity of the Cult of Mithra.

At this point I'll get into some specific examples. First consider the Kingdom of Armenia. Being both a once-independant kingdom sandwiched between the two empires as well as the guardian of the legendary "Gates of Alexander", a mountain pass that was a key access point for steppe nomads looking to raid both empires, Armenia was naturally greatly contested between the two empires.

The militarization of Persian culture extended west into Armenia, which Romans and Persians fought over but which remained partly independent and culturally separate. The Armenians converted to Christianity in the fourth century, which separated them further from the Persians, who were Zoroastrian for the most part (although with sizeable Jewish and Christian minorities, and also local traditional religions).

Christopher Wickham, "The Inheritance of Rome"

Here we can see a region that, while under Sassanid control, still chose to look Westward towards the Christian religion, largely as a political tool to defy the Sassanian Shahanshah. In fact, not only were they a Christian kingdom before the conversion of the Roman empire, but their Christianity went back to the time when the faith was still a persecuted religion in Rome. However, while their faith was initially used as a means of forging an independent identity apart from both Roman Paganism and Sassanian Zoroastrianism, by the sixth century, Armenia had become one of many Christian regions in the border zone, largely indistinguishable from those whose conversion was brought on because of, rather than despite, Roman political power.

Another one of these Christian regions was Edessa. One of the great holy cities of the Byzantine Empire, it spent the first third of the seventh century changing hands like it made a bet with the robot devil. Starting off under Byzantine control, in 609 B.C. it was taken by the Sassanians before once again being reclaimed for Rome by the Emperor Heraclius; then, in 638, it fell to the Caliphate.

Edessa lay on the fault-line between the rival empires of Rome and Persia... Edessa was famed among Christians everywhere as the “Blessed City.” Christ Himself, so it was said, had penned a missive to one of her kings... No wonder, given these indubitable marks of divine favour, that Christians should have believed that Edessa was destined never to fall. Certainly, the protective sympathy extended by her rulers towards the Church had enabled the city to become a veritable hot-house of Christian scholarship. The hymns, and the prayers, and the translations written there would serve to make Syriac, the language spoken by the Edessans, the lingua franca of the entire Christian Near East.

Tom Holland, "In the Shadow of the Sword"

While Edessa was one of many largely Christian cities to spend at least some time under Sassanid rule, as Holland points out, one thing that makes it stand out from the rest is its strong tradition of religious scholarship. However, Edessa was not just home to a large tradition of Christian scholars, but, additionally, had a significant number of Jewish scholarly institutions as well. While Judaism was tolerated with gritted teeth by Roman and Byzantine authorities, the Chosen people had fared much better under the benign apathy of the Shahanshah. Perhaps because of this, one of the most interesting facets of the border region between Byzantium and Iranshahr was the curious ways in which Judaism and Christianity co-mingled. Again, quoting Holland:

In Mesopotamia, for instance, Christians still refused to eat meat that had not first been drained of all its blood, exactly as prescribed by the Torah. They celebrated the resurrection of Christ on a date arrived at by a specifically Jewish method of calculation. Even the title by which they habitually addressed their priests—rabban—sounded similar to, and had the same meaning as, rabbi. Granted, none of this implied any sense of kinship between Jewish and Christian leaders. Quite the contrary: the rabbis’ and priests’ consciousness of what they shared only made them all the more determined to draw rigid dividing-lines between their two faiths. Yet, in reality, those dividing-lines were often blurred. In Edessa, for instance, the mutual obsession of Jewish and Christian scholars with one another’s labours bore a particularly spectacular fruit: for the first translation of the Old Testament into Syriac was most likely made, not by Christians at all, but by Jews, who then subsequently converted and made a gift of their work to the Church.

Most interesting, however (to me at least), is that on the border regions, we get one of the few points in history (exempting the first) where there is Judaeo-Christian syncretism:

To many Jews, it was not always apparent that an acceptance of Jesus as the Messiah did necessarily require them to stop being Jewish. Likewise, among the ranks of Christians, there were many who persisted in obeying the Torah and who regarded Paul, not as a saint, but rather as “a renegade from the Law”—the heretic of heretics. Some, indeed, saw no contradiction in hedging their bets even more comprehensively by invoking both the One God of the Jews and the Trinity of the Christians simultaneously.

To me, this above all else is what stands out about the liminal zones between the "two eyes of the world". In the sixth and early seventh century A.D., ones sense of identity sprang much more from your religion than from the empire that you paid taxes to. What makes the border regions in particular interesting is that you see much more dialogue between cultures and traditions that otherwise tended to distance themselves from each other.