They would probably be better off establishing client states in that region, possibly booting out unruly natives in Persia and sending them to Bactria. Then give the land to Latins or Greeks.
If it was this easy, the seleucids would have done this a long time ago.
That requires a foresight, ambition, and need that simply was not present and never would be present. The empire has no need to expand this way at all, and no need to arbitrarily divide it up into 3.Thats why you Split it into a Northern one sort of like the Frankish Empire under Charlemagne, a Southern one which would be the Quiet territories of Hispania, Africa, Aegyptus, Greece, Illyria, Anatolia, Syria, and so on you get the Picture. And then you have an Empire in Persia.
Rome doesn't need to arbitrarily expand into functionally useless territory to survive. Why does Rome need the mountainous and hard to control and manage areas of Persia when they could already have access to the Indian ocean, and thus bypass the land trade entirely, by controlling Mesopotamia?Or you could do it differently but what I'm mainly getting at is if Rome is to survive Rome is going to have to branch out and change.
That would never happen. Not only does it make no economic sense-access to the Mediterranean afforded access to an incredibly lucrative market-for that reason it was the principal goal of every civilization centered in the middle east going back to the Achaemenids at least, and its value was not lost on the Assyrians and Babylonians either. Seleucus went to great lengths to control Syria, which was just a rural desert with few urban centers to speak of before he built it up, just to have access to the Mediterranean. The Parthians and the Sassanians, had they possessed the military capability to do so, would have liked nothing more than to have a Mediterranean port.This Persian Roman Empire would most likely not have any holdings in the Mediterranean sea