Though the Umayyad period was in a constant flux, with governors and officials changing all the time, making a prolonged programme of discrimination on a local level uncoordinated and rare, its not particularly hard to find plenty of instances of oppression by the Arab conquerors, much of it cultural. The Umayyad system was, at its core, inherently pro-Arab and pro-Muslim. Non-Muslims were excluded from a variety of positions (though exceptions to this were surprisingly common) and faced harsh taxes, however conversion for Non-Arabs was a humiliating ordeal. Non-Muslim converts would have to attach themselves as a client to a local Arab tribe, and if accepted would be forced to change their surname and accept an extremely low status in this tribe. Persian nobles belonging to ancient lineages with a proud culture can't have been happy about being forced to endure this humiliation just to be accepted into civilised society. Indeed many didn't even bother, according to Bulliet a majority of Iranians clung to their pre-Islamic faiths all the way through the Umayyad period, with Islam only becoming dominant in Iran sometime in the early 9th century - 200 years after the conquest! Umayyad infrastructure was inherently pro-Arab. Persians faced, under this system, serious discrimination which was also faced by Greeks, Turks, Berbers and the many other nations which dwelled in the Caliphate.

Besides the ingrained biases in the Umayyad system, there were also specific instances of discrimination targeted at Iranians. Some dubious hadiths can be found reported in the Umayyad era which have clearly been fabricated just to discredit Iranians. There are two instances of fabricated anti-Iranian hadith which I can recall, one in which Abu Bakr heard two men conversing in Persian and demanded they stop, claiming that Persian was the tongue of the lazy and sinful (This hadith is poorly corroborated and its unlikely Abu Bakr was ever even with any Persian speakers) and another in which Muhammad allegedly claimed that passing a cup around a circle made the devil smile, which is an attack on traditional Persian drinking practices, and patently ridiculous given Muhammad likely didn't even know of such practices. The circulation of these hadiths demonstrates that anti-Persian sentiment was high among Muslims at the time and Islam was often used as an excuse to oppress Persian culture.

Similarly, the Arabs made several attempts to enforce Arabic as the official language of the caliphate starting with Abd al-Malik ibn Marwan. This really grated Iranians given so much of the civil service was based off of the Persian system and now had to be conducted in an alien language. There are some reports of Arab expeditions specifically against Persian speakers or Zoroastrians but they are hard to corroborate and often only appear written down by much later, Persian historians. Even if there were some governors who persecuted Iranians, most governors didn't last long enough to get much done in the Umayyad period. Whatever the severity of the enforcement of Arabic as the official language, the very fact that it was the official language was seen to be extremely unfair by Persians.

Of course, within a century all these oppressed minorities revolted and started a new caliphate which reversed most of these discriminatory practices. Infact, a strong faction emerged in the court known as Shu'ubiyya which were proto-Persian nationalists dedicated to the suppression of Arab culture and the promotion of Persian as a superior culture and ethnicity. By this time there was even talk of making Persian a co-official language in which the Quran could be read alongside Arabic but it never materialised as far as I remember, though the Quran was unofficially translated into Persian several times.

So all this discrimination didn't really last very long and it was sporadic and often poorly enforced, but it was certainly there and certainly extensive; after all, the minorities of the Caliphate fought a bloody war just to reverse it, so it had to be fairly bad!