It doesn't seem like the Roman Persia wars would have stopped without Islam, as each side proved rather incapable of completely subjugating the other. This went on for centuries. For how many more centuries could the situation remain like that? Would Turkish horded eventually take Persia anyway?
I see what you mean.
The Roman (Byzantine) - Parthia (Persian) wars went on for six centuries or so. In the world without Islam we have these empires staying and whence your question.
My opinion that some things are going to change.
Even during the last
Byzantine–Sasanian War of 602–628 the outcome of this war was seriously (if not decisively) influenced by the Western Turkic Khaganate.
So we already have the third really
big independent player in the traditional game of two (Rome vs Iran).
And that 'third big guy' stays in ATL - Khazaria as a successor state of the Gokturks. It stayed in OTL but was seriously crippled by the Arabs.
Here without Caliphate we would have one more "great world power" in addition to the Roman Empire, Persian Empire and China. And that Empire would stay for good.
And that is exactly in between the Byzantines and the Persians.
The Khazar State would not be a "horde" of the mindless barbarians, that would be even more sophisticated Empire than it was in OTL, as in this ATL it would probably include territories to the South of the Caucasus.
That would be an interesting game.
The point here is - if the two players ally against the third player, this third player is actually in great danger. But it is not certain that "this third player" destined to loose is necessarily Persia.
The Khazar Khagan and Persian Shahanshah might unite against the Roman (Byzantine) Emperor. Khazaria getting Anatolia and Persia getting Syria and Egypt.