Medieval Muslim scholars preserved many classical Greek source works, but apparently not many classical Persian ones. Did they have a Greek bias, or was there a lack of Persian sources to begin with?

The suspicion is that there was a lack of Persian sources to begin with. Also you have to deal with the question of what you mean by "classical" Persian.

The problem of the Persian kingdoms (and by extension, the success of the Roman state), is that in the Persian dynastic system, an overthrow of the ruling dynasty necessitates a change in the identity of the state. This is why we think of the Achaemenids, the Seleucids, the Parthians, and the Sasanids as different entities.

Compare this with the changeover from the Roman Republic to the principate of the Julio-Claudians, then the changeover from dynasties to the Vespasian and the Nervan-Antonine, and onward to the dominate and the Christian emperors. In the Roman world, a change in dynasties did not involve a change in state identity, EVEN if structurally, new dynasties were completely different from older ones (compare Tetrarchy with Severan).

This problem of state identity has an impact on the desirability of "classical" writing, as (from a marxian perspective) a means of legitimating the current state.

This is why whatever extant (or claimed) continuity in Persian literature tends to be religious, because the essential divine legitimation from Zoroastrianism WAS needed for continued operation.

However, for non-religious writing, any literature tied to the older regime, especially state sponsored ones that weren't technically beneficial (i.e military or medical manuals) ceased having utility because they legitimized the wrong state.

We can see the destruction of older "state" documentation in the fact that within the continuous Persian literary tradition of themselves, as embodied in the 11th century Shahnameh (Book of Kings), the oldest historical figure they can remember is Darius III, the last Achaemenid king, and ONLY because he was defeated by Alexander the Great, who is also in the Shahnameh. All the kings before Darius III, like Cyrus or Xerxes, they had little clue about. In fact, from the scholarship, it seems the Sasanids learned the most about their Achaemenid past through their interactions with the Romans, not via surviving literature which bridged the seleucid-arsacid gap.

Compare this to the roman world, where the flexibility of their governance allows for them to incorporate a classical corpus not even of their own cultural origins, i.e. Greek or Judeo-Christian. (The only other similar "civilization" which does this is China, which also arguably supports the case for literature longevity via state-accepted appropriation despite government turnover.)

So by the time the Arab Conquests took hold in Persia and the Mediterranean, Muslim scholars in Persia were dealing with, at best, only surviving Sasanid literature, which itself was ruptured from the Persian states before it. Whereas in the former Roman mediterranean, Muslim scholars could work with a world that was actively continuing pre-Roman scholarship.

It also helped that there was an active Syriac-speaking population which had been busy for some time copying Greek works into a semitic language close enough to Arabic to be generally understandable (with the usual caveats regarding intelligibility), and that the center of the early caliphate in Damascus maintained Greek administration.

Persian, of course, is Indo-European, and for more than 100 years after the conquest was not the center of power. That's plenty of time for the fairly crushed Sasanid state to have a literary survival and an identity crisis (it only took Britain 60 years to almost completely changeover identity after Roman withdrawal), combined with the fact that Persian was further from Arabic than Syriac was.

tl;dr - lack of Persian sources, due to the cultural structuring of the Persian states when it comes to government turnovers. Greek was more available, more accepting of older ideologies, and already translated into a closer language to Arabic.