Middle Persian followed by New Persian
Name
"Middle Iranian" is the name given to the middle stage of development of the numerous Iranian languages and dialects.: 1  The middle stage of the Iranian languages begins around 450 BCE and ends around 650 CE. One of those Middle Iranian languages is Middle Persian, i.e. the middle stage of the language of the Persians, an Iranian people of Persia proper, which lies in the south-western highlands on the border with Babylonia. The Persians called their language Parsik, meaning "Persian".
Another Middle Iranian language was Parthian, i.e. the language of the northwestern Iranian peoples of Parthia proper, which lies along the southern/south-eastern edge of the Caspian sea and is adjacent to the boundary between western and eastern Iranian languages. The Parthians called their language Parthawik, meaning "Parthian". Via regular sound changes Parthawik became Pahlawik, from which the word 'Pahlavi' eventually evolved.  The -ik in parsik and parthawik was a regular Middle Iranian appurtenant suffix for "pertaining to". The New Persian equivalent of -ik is -i.
When the Arsacids (who were Parthians) came to power in the 3rd-century BCE, they inherited the use of written Greek (from the successors of Alexander the Great) as the language of government. Under the cultural influence of the Greeks (Hellenization), some Middle Iranian languages, such as Bactrian, also had begun to be written in Greek script. But yet other Middle Iranian languages began to be written in a script derived from Aramaic. This occurred primarily because written Aramaic had previously been the written language of government of the former Achaemenids, and the government scribes had carried that practice all over the empire. This practice had led to others adopting Imperial Aramaic as the language of communications, both between Iranians and non-Iranians, as well as between Iranians.: 1251–1253  The transition from Imperial Aramaic to Middle Iranian took place very slowly, with a slow increase of more and more Iranian words so that Aramaic with Iranian elements gradually changed into Iranian with Aramaic elements.: 1151  Under Arsacid hegemony, this Aramaic-derived writing system for Iranian languages came to be associated with the Parthians in particular (it may have originated in the Parthian chancellories: 1151 ), and thus the writing system came to be called pahlavi "Parthian" too.: 33 Aside from Parthian, Aramaic-derived writing was adopted for at least four other Middle Iranian languages, one of which was Middle Persian. In the 3rd-century CE, the Parthian Arsacids were overthrown by the Sassanids, who were natives of the south-west and thus spoke Middle Persian as their native language. Under Sassanid hegemony, the Middle Persian language became a prestige dialect and thus also came to be used by non-Persian Iranians. In the 7th-century, the Sassanids were overthrown by the Arabs. Under Arab influence, Iranian languages began to be written in Arabic script (adapted to Iranian phonology), while Middle Persian began to rapidly evolve into New Persian and the name parsik became Arabicized farsi. Not all Iranians were comfortable with these Arabic-influenced developments, in particular, members of the literate elite, which in Sassanid times consisted primarily of Zoroastrian priests. Those former elites vigorously rejected what they perceived as 'Un-Iranian', and continued to use the "old" language (i.e. Middle Persian) and Aramaic-derived writing system.: 33  In time, the name of the writing system, pahlavi "Parthian", began to be applied to the "old" Middle Persian language as well, thus distinguishing it from the "new" language, farsi.: 32–33  Consequently, 'pahlavi' came to denote the particularly Zoroastrian, exclusively written, late form of Middle Persian. Since almost all surviving Middle Persian literature is in this particular late form of exclusively written Zoroastrian Middle Persian, in popular imagination the term 'Pahlavi' became synonymous with Middle Persian itself.
The ISO 639 language code for Middle Persian is pal, which reflects the post-Sasanian era use of the term Pahlavi to refer to the language and not only the script.Transition to New Persian
The modern-day descendants of Middle Persian are New Persian and Luri. The changes between late Middle and Early New Persian were very gradual, and in the 10th-11th centuries, Middle Persian texts were still intelligible to speakers of Early New Persian.  However, there are definite differences that had taken place already by the 10th century: