Touraj Daryaee's Sasanian Persia does not mention this particular incident, but the timing would mean that it was some time after the sack of Jerusalem in 614 (and the taking of the "True Cross") but before the conquest of Egypt, Southern Syria or Anatolia. There's good reason to doubt that actual conversion was demanded. For one, the Sasanians had by then accepted Armenian Christianity for nearly a century and a half (though only as the result of needing Armenian support in a brief civil war). For another, Zoroastrianism is unlikely to have been perceived as a universalist religion by this point. Our record for religious policy in the latter half of the empire is spotty or complicated by sketchy textual histories, but it's still usually thought that the tradition received in Middle Persian texts (e.g. the Denkard) had ossified by this point.

How can we then understand this supposed demand? I think we must first consider the pivotal role religion played in this war. The sack of Jerusalem is the most famous incident, of course, but even more important was Heraclius' retaliator sack of Ganzak in 624, which was probably the final straw undermining Sasanian nobility's confidence in Khusrau. There's a received history of Heraclius' majestic counteroffensive, but much suggests that internal Sasanian power struggles and war fatigue was more to blame. Even the quite Heraclius-sympathetic account of Theophanes does not maintain that Heraclius actually defeated Khusrau's main army at Ganzak, but that Khusrau fled and that it "scattered" following a skirmish with some Arab scouts. This "official" account never made much sense to me, nor Khusrau's subsequent attempts to rally troops for the defense of Ctesiphon while Heraclius opted against pursuing him, and I suspect Khusrau had his hand forced by internal power struggles.

Regardless, the role of religious ideology is clear in the course of Heraclius' campaign. Now, I can only really speculate in what the basis for this rumor of a demand for conversion was, if there was any. One, it is possible that Khusrau may have demanded Heraclius to resign his religious authority and/or elevate the status of the Patriarch in Ctesiphon, to increase Sasanian clout over Christians and make it easier to rule Syria. Two, it is possible that Khusrau may have demanded the formal submission of the Roman Emperor to the Sasanians. This seems particularly likely when we take into account the Sasanian tradition attested in royal inscriptions going back to Shapur I, of considering the Roman Empire a vassal state, by virtue of the massive tribute it paid. Moreover, given that the King of Kings was also the supreme authority of the Zoroastrian religion, there is obviously a connection there, of essentially demanding the submission of Christendom itself.

The third possibility is that it is simply a Roman extrapolation from Khusrau's humiliating sack of Jersualem. If he had taken the true cross, what else was he going to do? This may have been propaganda issued by the priesthood or Heraclius himself to raise morale for what was sold as a holy war (Daryaee even describes it as the true "first crusade").

Whichever of these is true, from internal policy toward Christians, the Sasanians of the post-Kerdir era (late 3rd century) were evidently concerned with political-ideological submission way ahead of demanding conversion. The concern with demanding Armenian conversion had a strong cultural element and also an aspect of political defiance on the part of the Armenian nobility, and even there, Chrisitanity was eventually accepted.