MIDDLE IRANIAN language includes the Iranian dialects as they appear from about 300 B.C. to about 900 A.D. They are in general called Pahlavi, which is only the regular development of a derivative of the OP word 'Parthian'. It is clearer to discuss the dialects partly by dialects and partly by the extant remains.
I. Arsacid Pahlavi was the official language of the Arsacid dynasty of Parthi4, which ruled from 250 B.C. to 226 A.D.; it did not die out with the dynasty. It is represented in some bilingual inscriptions alongside the Sasanian Pahlavi, where it is often called Chaldaeo-Pahlavi or Parthian; by the parchment manuscripts of Auroman; and by certain Manichaean texts from Turfan (IV). It is also called Northwest Pahlavi, and apparently was developed from a dialect which was almost or quite identical with that of Media.
II. The Sasanian or Southwest Pahlavi was the official language of the Sasanian dynasty, which ruled from 226 A.D. until the Mohammedan conquest in 652. It is known from some rock-inscriptions of the kings in the general region of Persepolis, datable in the 3d and 4th centuries, some being accompanied by a translation into Arsacid Pahlavi or even by a second translation into Greek; from some texts on Egyptian papyri, of about the 8th century; from many religious texts preserved by the Zoroastrians (III); and from some of the Manichaean texts found at Turfan. In inscriptional form it can be observed in legends on coins, seals, and gems, until near the end of the 7th century. It appears to have developed from Old Persian or from a very similar dialect.
III. The `Book-Pahlavi' includes the writings preserved by the Zoroastrians of Persia and India, forming a very considerable body of literature divisible into (1) translations of parts of the Avesta, with commentary, (2) texts on other religious subjects, (3) texts on other than religious topics. They represent both Sasanian and Arsacid Pahlavi. They are written in an alphabet derived from that of Aramaic, and, like all the early Pahlavi writings and inscriptions, contain an extremely high percentage of Semitic words; but many of these were to be read with the Iranian equivalents, even as we write id est and say `that is', viz. and say `namely'.
IV. The manuscripts found at Turfan, in the early years of the 20th century, give us texts that are mostly of the 8th and 9th centuries, though some of them go back almost to the beginning of the Christian era. These texts represent several
dialects, including the Arsacid and the Sasanian types, the Sogdian (known also from a trilingual inscription of Kara-Balgassun), and a dialect known as `Eastern Iranian', perhaps a derivative of northeastern Scythian, in which there are texts of the Buddhists of Khotan. The notable peculiarity of these Turfan texts is that they are written in relatively pure Iranian, without the Semitic writings for the words which are to be spoken by the Iranian equivalent.
V. Among the earliest traces of Pahlavi, however, are certain legends in Greek characters on coins of Indo-Scythic rulers of the Turuka dynasty in northwestern India, belonging to the first two Christian centuries.