I'll speak mainly within the purview of Al-Tabari's texts and the Abbasid Dynasty as that is within my.area of familiarity.

To understand why the Persians had such an influence on the Abbasids we can start at the placement of the new capital of the Caliphate in Baghdad. Given its close proximity near the former Sassanid capital, this hints that at least geographically, the Abbasids had grown to rely on Persian bureaucracy and governance as a model. Furthermore, some theorize (though Al-Tabari highly suggests this) that the Abbasid connection to Persian rule was a way to cement legitimacy beyond the immediate area. Notably we see a large sponsorship of scientific and literary works from the Abbasids that were not written or translated to not only Arabic, but also Persian. Similar to the Byzantines at the time who continued the tradition of writing and producing literature in Greek, its possible that much of the Persian influence that the Abbasids borrowed from were products of a teleological justification.

Literary continuity commentaries aside, quite a lot of works that comes from the Abbasid Caliphate ended up being written in Persian and Arabic and the large projects translation of Greek and Roman philosophical works resulted in both Arabic and Persian transcriptions. Perhaps more pertinently, the political rise of the Abbasids depended on cooperation of the Persians against the Umayyads and though there was political friction afterwards, the prevalence of Persian political figures and people within the Abbasid Caliphate resulted in a much more inclusive rule.

I'm fairly certain there are many aspects I've missed, so if anyone has any addons, feel free to add addendums.

Persians dramatically outnumbered Arabs. They were vastly wealthier, had a tremendously larger cultural tradition, including literacy, and they were more urban and sophisticated. The Umayyads tried hard to keep the Arabs ruling the Near East segregated from local populations in order to maintain Arab cultural unity, but in the far-flung areas of Khorasan (what is today Uzbekistan) this proved impossible, and Arabs and Persians mingled a great deal. The Persians had more wealth and culture, so they tended to make the Arabs more Persian than the converse.

A good example of this is when al-Mansour built Baghdad in the 760s. The city was arranged in such a way as to conform to Persian obsessions with astrology, and when Mansour was installed, he had it done in the manner of a Persian king.

Let me start of with a bit of a generalization, and then I'll try my best to support it. The society of the Arabian Peninsula was one that was constituted of a a largely oral cultural tradition, governed with tribal affiliation rather than the bureaucracy and politics we are familiar with, and without the same sheer infrastructure that we see in the Byzantine, Sassanid, and Old Roman Empires. This means that when they powerful Arab armies took over lands outside the Arab peninsula, they began to borrow from them on a cultural level, and administrative level, and on an architectural/structural level. They would be quite impressed by the way that foreign emperors would live, the massive architecture of these different civilizations, and they would find a great deal of use in the bureaucratic structures that helped other empires govern their multiethnic domains (much like the Arabs were, as you can't structure your politics around tribal relations completely when there are people that don't fit into that tribal mosaic.). I am not much of an Art Historian, or one too well versed in Architecture, so I will focus on the administrative/political level (art and architecture is important, and I could talk about it, but I only know about it inasmuch as I can relate it to the political and administrative so I don't really want to start taking authority in that area.)

So the first work I'd like to introduce you to is "The Administration of Abbasids Caliphate: A Fateful Change in the Muslim History" by Mohammed Tahir, a great Pakistani academic, in The Pakistan Journal of Commerce and Social Sciences. In the beginning of his article, he actually conceptualizes the defeat of the Umayyads and the Rise of the Abbasids, as well as the change of capital from Damascus to Baghdad as a battle of influences. The Capital had already been moved from Medina to Damascus, because of the centralization, and in a sense to also remove the Capital from its semi-nomadic roots in the desert to the high culture of the Byzantine/Levantine cultural sphere, though they retained a largely Arab-Centric Society and leadership. He argues that much of the ways that the Caliphs would begin to present themselves and operate during the Umayyad Caliphate was influenced by the way the Byzantine emperor operated, and it greatly influenced the more political nature of the Caliph ul-Islam. This influence would be changed when the Abbasids would adopt a far more Persianised point of view. They would move the capital towards Baghdad, and create a sort of Second person capital, and a new more persian empire on the remains of the more Arab Kingdom that they had vanquished to the far ends of Al-Andalus. (Although to be fair the Capital was actually moved to Kufa, then to Baghdad, then it had a stint in Ar-Raqqah and Samarra for short periods in between being in Baghdad as well). The Abbasids used the discontent of the Arabic centric politics that didn't really work with the growing diversity of the empire, to take control of it -- much of this misbalance was owed to what was effectively the swallowing of Persia and in turn its people and their culture.

It is also important to consider linguistic politics here as well. The Persian language, unlike many others in the region, took on a really important role. The language adopted an arabic writing system, and there was actually a great deal of patronage for works in the Persian language -- entire bodies of work and poetry in the Persian language flourished and the language itself would interact a great deal with Arabic and help create Arabic vernaculars. Much of these details can be discussed in Hayrettin Yucesoy's work "Language of Empire: Politics of Arabic and Persian in the Abbasid World.", he is a professor of the Medieval Middle East at WUStL.

The Abbasids would employ Persian methods of administration and bureaucracy, and even had many persians occupy important positions in government, marking a movement away from the ethnic rulership of the Arabs over all others (actually one of the Reasons the Umayyad fell in the first place). They also would adopt, from the Sassanid emperor, the idea of a Caliph who ruled with the divine sanctioning of God, a concept that would forever mark the position. The court life of the Caliphate would look much like that of the Persians, and the Caliph would wear a black turban, carry a staff of the prophet, and have the Quran before him on his throne as every one of his attendants and officials would kiss his hand when greeting him. They borrowed their system of investiture from the Persians as well.

They would also institute the Wazirate, a Wazir being a sort of prime minister, someone invested with all the sovereign powers to help the Caliph rule. They were often Persian, and would help in this transformation of administration. In addition other offices, such as that of an Executioner, and a diplomatic corps. The position of Wazir would exist and die with the Abbasid caliphate. So to conclude, the Abbasid Caliphate was emerged in a way as a more persianised alternative to the Umayyads, and they would go onto use Persian administration and language (and also artistic culture that I hope someone will be able to discuss more than I could.) I hope that gives you some insight, and if you'd like to know more, feel free to ask!

An interesting read on this issue, mainly is about women's position in Islamic societies but gives quite a good idea about Persian (actually, Mesopotamian) influence over Islam and Islamic states: http://arizona.openrepository.com/arizona/bitstream/10150/291738/1/azu_td_1353097_sip1_m.pdf