They can... sort of... use the Euphrates and Tigris. You still have a long way to transport goods from any Mediterranean port, and then any army at the Zagros will require more overland transport at the other end too.
It's not impossible, I mean, Rome has temporarily held the area and the Ottomans did for a long time too. I think these examples, coupled with oh-so-many polities holding Iran-Mesopotamia show that Rome could only hold it against a weak Persia. If the area is under a vigorous and dynamic power, they can simply project force into the area more easily than Rome can.
I suspect this is the strongest method for defence in the west of any Roman/Byzantine Empire. Holding the Adriatic with Venetia as an impervious HQ and logistics centre makes it much easier for the Romans to project force into the region. Couple that was the mountainous terrain, and it's a reasonably "natural" border for Rome.
It does require Rome never falling too far behind, however. It's reasonably easy to imagine Rome stumbling when the Mediterranean trade shifts to the Atlantic and disrupts its economy. If Rome is going through a period of turmoil and the rising powers in France (or Germany) sense weakness, there isn't much stopping them once they penetrate the mountain passes.