There are a couple reasons, but by far the most salient is the utter destruction of the Sassanid state very soon after that apocalyptic war concluded.

This state of affairs was far from inevitable, but I think it is exceedingly likely that the postwar period would have been a minor dark age and time of chaos for the Persians even had Khalid ibn Al-Walid and his fellow warlords not shattered the Persians at Qadisiyah and then again at Nehavend. Attempting to trace Persian royal politics immediately after the treaty with Herakleios is a little like reading a condensed version of the Roman Third Century Crisis — murder, coup after coup, and even plague. Even candidates who the modern reader (and, indeed, the contemporary political observer) would think to be strong contenders last mere months. Shahrbaraz, the conqueror of so many Byzantine armies and provinces, was one of these — even his immense prestige did not save him. In happier times he could perhaps have been a Persian Charles de Gaulle, restoring prestige and self-regard in a defeated but still-proud nation.

This chaotic succession was essentially an obverse of the coin vis a vis Byzantium after Maurice’s murder; the sudden disappearance of royal legitimacy opened to floodgates to all contenders. It was not a period conducive to coherent native Persian written accounts, to put it mildly. On top of everything else, scholars believe Persia and Byzantium were both being ravaged by a recurrence of the Justinianic bubonic plague — which, in addition to destroying many Persian claimants to the throne, goes far in explaining why the once populous and rich Persian Empire could not hold back the Arab tide.

Even the return of stability under Yazdegerd III was not enough to reverse this trend, though had Muhammad never been born it is very possible that it may have eventually led to a return to normalcy. Yazdegerd had competent counselors and courageous and skilled generals — though had the Arabs not arrived, the weakening of the imperial throne when compared to the increasingly powerful magnates of the provinces would have been a problem in its own right.

Rostam Farrokhzad’s stand against the Arab armies was famous enough to make it into Arab sources, and it is worth noting that the Persian armies, weakened though they were, managed a remarkable tactical victory over the Arab armies at the Battle of the Bridge — which Herakleios, for all his acumen, was utterly unable to equal. There are also hints in the sources that the Persian cavalry was conducting a skilled and effective guerilla campaign in occupied Iraq against the Arabs. It is easy to imagine these exploits being part of a heroic Persian court history, culminating in a grand battle where Sassanid valor prevailed over the invaders — but of course, it was not to be.

What really prevented a Persian resurgence was, in a word, pride. Yazdegerd and his generals could have decided to fall back behind the Zagros mountains, much as Herakleios and his counselors decided to fortify the Taurus range — the guerilla war Persia was waging could easily have taken on the tenor of the famous Byzantine Theme system, but even had Yazdegerd been astute enough to grasp this his great nobles were thirsty for a battle. I suspect that the Persian magnates had been deeply wounded by their defeat at Herakleios’ hands, and a great victory would have gone a long way towards restoring Persian prestige. Herakleios, as a victor, had enough legitimacy to force through what many Byzantines saw as a dishonorable policy; Yazdegerd, his throne creaking beneath him, had no such luxury.

The rest is well-known; Persia’s cataphracts were outmaneuvered and destroyed, despite a desperate resistance. And while Persia’s fight for independence made it into later Persian odes, poetry and history, the war against Byzantium was forgotten in the chaos of postwar Persia.

Some sources: In God’s Path, by Robert Hoyland (use with caution; it is too hard on Persia and Byzantium, but is nevertheless a good source); The Restoration of Rome, by Peter Heather; A History of the Byzantine State and Society, by Warren Treadgold; and In the Shadow of the Sword, by Tom Holland (a good introductory read; well written and dramatic, but refer to Treadgold for a more analytical take).