I'm not sure if it was ever "really close", but I don't know enough about the conditions of the siege of Constantinople.
Where's BG when we need him . . .
And even if it did work, I'm not sure it has any "long term". It grossly overstrained the Sassanid state.
Here I am!
The Iranians undoubtedly could have won the "great war" with Rome in about 615, and got significant concessions out of it- iirc, one of Heraclius' desperate offers for peace involved giving up all territories east of the Euphrates, all of Armenia, and paying a massive sum of money to Ctesiphon indefinitely. It was the pig-headedness of Khusrau II, who let success go to his head, that stopped this.
I don't think the attacks on Anatolia, Cappadocia apart, were ever really much more than glorified raids- certainly the Iranian armies that reached Chalcedon were only outriders. Heraclius was able to train his armies in Anatolia in the 610s, after all, which would not have been possible had the whole region been under Iranian control. There was undoubtedly a lot of devastation (archaeology at Sardis shows the ancient city ending with a layer of burnt material in the 610s, for example), but not on the level of a permanent conquest.
Constantinople I would say is probably more-or-less out of the question for the Avar/Iranian alliance to capture, for as long as the Romans maintain their sea-power. Even then, there's a high probability of failure. The Arabs brought significant fleets to the City in the 670s and 710s, and their attempts ended in failure, after all.
So, all in all, no, I don't think a Persian conquest of Rome at any point is doable- I think the seventh century war is as close as they could get, and their success there is pretty extraordinary. Lest I be accused of bias, I should also state that I think a Roman conquest of the Iranian plateau is just as unlikely. Mesopotamia is probably
just within the bounds of possibility for Rome to take and hold, at least for a few generations. But beyond, nah, I can't see it!