I think there are two major reasons why Persia was able to retain its language as well as much of its culture, while other regions that came under Arab rule could not, one is historical and the other is linguistic.

1. Historical: The first major Islamic empire was the Rashidun Caliphate which was founded in Medina in 632. Between 632 and 661, the empire expanded at a breathtaking pace and incorporated the Levant, Mesopotamia, Persia, Northern Egypt, Southeast Turkey and the Caucasus into its realm. Now the official policy of the Rashidun Caliphate was to allow the provinces to maintain their local languages and customs.

In 661, the Rashidun Caliphate was replaced by the Umayyad Caliphate which was notorious for its Arab favouritism. The Umayyads imposed Arabic as the official language in all provinces, often by force, and particularly in Persia whose resistance irked the governor of Iraq, Hajjaj ibn Yusuf. Islam at this time was tied to Arab ethnic identity and required formal association with an Arab tribe. This conflicting relationship between a religion that claimed universality and a political class that clearly favoured a particular ethnicity was one of the major reasons for the Abassid revolution (747-750) which overthrew the Umayyads, and in which many Persians and other ethnic groups took part.

The Abassid Caliphate was established in 750 and it was this empire that witnessed what eventually came to be known as the Golden Age of Islam. Since many Persians had taken part in the revolution that helped bring the Abassids to power, Persians found themselves in charge of many important political and bureaucratic positions. The Abassids heavily relied on Persian bureaucrats from the Barmakid family and this gave way to Persian customs being broadly adopted by the ruling elite. This process coincided with the rising number of Persians taking part in science, medicine, mathematics, philosophy and architecture. This led to the increasing importance of the Persian language, which would later go on to become the lingua francia of a great number of Persianiate empires.

2. Linguistic: Arabic is a Semitic language belonging to the Afro-Asiatic language family, while Persian is an Indo-Iranian language belonging to the Indo-European language family. You notice that most of the languages that Arabic replaced were linguistically close to it in terms of grammar and vocabulary, such as Syriac, which was a Semitic language spoken in modern day Iraq, Syria and the Levant, Palestinian Aramaic which was a Semitic language spoken in the Levant, and the Egyptian language which was an Afro-Asiatic language.

This linguistic familiarity helped in the wider dissemination and adoption of Arabic by various Semitic or Afro-Asiatic speaking provinces. But one should be careful to note that these languages weren't shiftly replaced by a single language, rather there was a slow and gradual process that eventually resulted in these populations speaking the same language, but many of the local dialects still retain much of the vocabulary of their ancestral languages.

Sources:

Cambridge History of Iran (Volume 4) by Richard Nelson Frye and Abdolhosein Zarrinkoub, Cambridge University Press, 1975

The Heritage of Persia by Richard Frye, Weidenfeld & Nicolson, 1962

From Aramaic to Arabic: The Languages of the Monasteries of Palestine in the Byzantine and Early Islamic Periods by Sidney H. Griffith, Dumbarton Oaks Papers, 1997