I imagine a cavalry heavy army is going to experience signficant problems crossing the Caucuses. Looking beyond that though, you still have the problem of what happens when you're marching around the Black Sea and Heraclius just takes his army and smashes through Anatolia and Asia. This isn't that far fetched, since he was able to turn the tide fairly rapidly IOTL. But even assuming he doesn't, well, then what? What happens when they reach Constantinople? The Avars were already besieging the landward side and making exactly 0 progress. A cavalry heavy Sassanian army that doesn't have any siege equipment isn't going to be able to take on the Theodosian Walls. And they aren't going to be able to starve the city into submission, since the Byzantines still rule the waves and can supply their capital with impunity.
The Sassanians can easily win the war, but they can't take Constantinople. However, winning the war requires Khosrau knowing when to quit. And it's kind of hard to accept a desperate peace offer from the Byzantines when it appears you are only a hairs breadth away from the entire empire collapsing and falling into your lap. If Khosrau quits while he's ahead, he can get some serious concessions. Of course, how long the Byzantines are willing to abide by those official concessions and how able the Sassanians are going to be able to defend them is another matter. Certainly an emperor who just ceded the most valuable half of his empire to the hated Persians isn't going to last long, so either Heraclius is going to be under immense pressure to renege on any treaty the moment Sassanian armies march home, or he'll be deposed in short order for another emperor who will renege on the agreement. And then your back in the same position, with a back and forth war that likely ends in only minor territorial changes, exhausting both empires.
It wasn't really a war that could be won by either side.
