Quite true. It wasn't just because of the jizya either; Islam at the time of the Umayyads was not a cosmopolitan religion. It was supposed to be for the still-tribal Arabs--foreigners just wouldn't "get it".

However, even though it was discouraged, Islam still did spread chiefly among the Iranian peoples, most importantly the nobility (I will explain why this is important later), who found understandably Islam to have much in common with Zoroastrianism, such as:

• Monotheistic [I know someone will get pedantic and say that Zorastrianism is technically dualistic but for our present purposes I don't think that is very important]

• Prayer 5 times a day

• Heaven and Hell (interesting aside: in modern Persian we can interchangeably use both the Zoroastrian and Islamic terms even today: pardiz [paradise] and doozakh are the Zoroastrian terms w/ indo-european roots, behesht and jahanam are the Islamic terms w/ semitic roots)

• Afterlife and judgement leading to end-of-time apocalyptic battle between good and evil

• Omnipotence, omniscience, and omnipresence of God

and many other things (Boyce's Zoroastrians: Their Religious Beliefs and Practices for more). The Arabs had already mad much exposure to Zoroastrianism, and the Iranians had already much exposure to Abrahamic religions--both Judaism and Christianity--and I think it is quite easy to see the cross-cultural influences between the religions.

I mentioned the nobility adopting Islam as important, this is because the nobility were also at the head of the Zoroastrian clergy. Whole noble families even converted (such as the Samanids), perhaps because they genuinely liked Islam, or perhaps because they wanted to keep their noble status. But with the gradual conversion of the nobility came the loss of influence of Zoroastrianism and the growing influence of Islam. It is also important to note that Zoroastrianism was not very uniform in practice--depending on where in Greater Iran you were, Zoroastrianism could very well be a mix of Indo-European semi-nomadic or tribal customs (Frye goes into this in great detail in The Golden Age of Persia) to where the Zoroastrianism practiced in one part of Iran could be almost like a totally different religion than in another.

So I mentioned the gradual conversion of the nobility to Islam--this started towards the end of the Umayyad reign. As more non-Arabs joined the ranks of Muslims, the more apparent the discrimination towards non-Arabs (or Ajam) became. This gradually led to the downfall of the Umayyads in place of the far more cosmopolitan Abbasids; in fact most of the Abbasid revolts were led by Iranians, most famously Abu Muslim Khorasani, who stormed Damascus and helped solidify Abbasid rule.

I hope this makes sense and was understandable because I haven't really proofread it lol, if you have any questions please ask. The main sources for this were Frye and Boyce.