This is an interesting question, but before we delve in to it, we have to talk a bit about terminology. Specifically, we have to define what "Persian" means. Do we accept a literal definition as in from the region of Persis? This makes some sense as the Achaemenid empire literally came from the region of Persis, and technically so did the Sassanian empire. But what about culture, can we say that these empires were culturally "Persian" (i.e. according to specific cultural traits that Herodotos and later Strabo elaborated on?). In the end, both definitions make life more difficult than necessary. You see, we have to impose a term to characterize both of these empires (things get even more fun when we add in the Arsakid empire) but that may ignore the surprising fluidity seen in the royal paradigm. For example, the Achaemenid empire would adopt numerous court practices from the Medians and the famous title "King of Kings" came from Mesopotamia.

For the Sassanians, many would characterize them as the Achaemenids "reborn" from the brutal Hellenophile Skythian Parthian interlude. This is not a good enough answer though, because if anything, the Sassanians were probably closer to the Arsakid Parthians (by the Gods do I hate this term!) than the Achaemenids. Militarily and culturally (not to mention politically) the Sassanians may be characterized as the "Parthians" reborn (a professor I talked to argued that the transition from the Arsakid to Sassanian dynasties was akin to the transition from the Roman Principate to the Dominate). Take into consideration that the Sassanians would rely--no sorry--would NEED the Parthian clans such as the Mihran, Karen, and the Suren to overthrow the Arsakids. Hell, apparently, the Surens actually supported the Arsakids initially before throwing their lot with the Sassanians. Even after their takeover, Ardashir was quick to marry Lady Myrod of the Arsakid house. All the signs just from these few elements strongly indicate the Sassanians (at least in their empire) wished to be construe their empire as the Parthian empire reborn.

But then, why do we call them Persians? I ignored a few important points that differentiated the Sassanians from the Arsakids. Elements such as zealous push for Zoroastrianism as well as putting Persis back as an important region cannot be underestimated. I think that this has more to do with geography than anything else though, being that the Sassanians originated from Persis(without going into too much detail, there is debate about the origins of the dynasty with some arguing that they were a Parthian clan themselves!) they may have used the homeland of a mighty empire as a means to legitimacy. It must be stressed that the Roman sources are really the big culprits for characterizing the Sassanians as "Persians." For this analysis, I must defer to Dr. Rahim Shayegan who argued quite convincingly that Roman historians wished to make themselves out as ancient Greeks fighting the terrible Persians. And just like in the late 400s BC, when the Greeks kicked Persian ass, the Romans would do it again! Clearly, these characterizations are problematic, but I must emphasize it is a bad idea to just start ripping into the Roman sources. In many cases, they provide important chronological information regarding the Sassanians, and their works cannot be dismissed, just carefully and critically analyzed.

Ok, now that you have most certainly fallen asleep, to your question: Were there any Persian Sassanian dynasts that converted to Christianity? Highly unlikely and this goes again back the Zoroastrian part of the royal ideology. And while it was not some terrible theocracy as argued by that idiot Terry Jones, the religious institution was an important part of state ideology as opposed to the Christian Romans. But wait a minute, didn't the Romans become Christian officially in the AD 370s? And the Sassanians arrived on the scene in the AD 220s, meaning there is a disconnect. Ah, but we casually ignored a state in the most unenviable position right between the Sassanians and Romans: the Armenians. This country was ruled by an offshoot of the Arsakid dynasty that had taken power in the AD 50s from a diplomatic settlement between Nero and Arsakid king Vologases I. What is interesting about this dynasty was that it converted to Christianity in the 300s AD, no doubt spurred to piss off the Sassanians! And think about it, if there was a dynasty that threatened the very existence of the Sassanians--it would be the Arsakids who could claim their old empire! Specifically, they could call upon the Parthians clans that had defected to the Sassanians. Now, I am not going to lie, this thesis is very controversial in that seeks to explain the rise of the Zoroastrian state apparatus in the Sassanian empire, but in my opinion it is a better explanation than a rather anachronistic explanation that the Sassanians wished to the restore Iranian (read=national) power in the Near East. I would rather trust arguments that explain events in a pragmatic and logical manner.

I hope this answers your question!

Sources used:

Decline and Fall of the Sasanian Empire: The Sasanian-Parthian Confederacy and the Arab Conquest of Iran-- Parvaneh Pourshariati

Arsacids and Sasanians, M. Rahim Shayegan.

EDIT: Grammer