The Byzantines did adopt some elements from Persian culture, in art, clothing, and even in warfare, but a wholesale "Persianization" stumbles on the political rivalry between the two: Rome/Byzantium perceived Persia as the main "other" power and civilization in the known world, and hence as alien and to a degree antithetical to themselves. As long as Persia remained an independent major power and foe, to adopt Persian culture wholesale and renounce their own Greco-Roman legacy would be tantamount to acknowledging Persian supremacy. It's no coincidence that their relations have been likened to US-Soviet rivalry.
Persian cultural traits could be increased the easy way if the wider Middle Eastern culture becomes more dominant over the Hellenistic tradition already in the Roman Empire, so that when a split occurs, the East will already be more "persianized". Otherwise, a conquest of Persia or at least a major part of it (preferably with a vassalization of the rest), which would result in the rise of people of Iranian origin to positions of power at court, would have the same result over time. However, even then, outright "persianization" is unlikely, and a fusion between the two cultures would be the probable result. If the "Roman" tradition continues strong, Constantinople will look to the West as much as the East, otherwise a mainly Asian state could develop along the lines of the Achaemenids.