The Middle Persian script developed from the Aramaic script and became the official script of the Sassanian empire (224-651 AD). It changed little during the time it was in use, but around the 5th century AD, it spawned a number of new scripts, including the Psalter and Avestan scripts.
There were a number of different versions of the Middle Persian script:
The inscriptional version of the script appears in inscriptions in clay fragments dating from the reign of Mithridates I (171-38 BC), and only in coin and rock inscriptions. In this version the script has 19 letters which are not joined together
The name Psalter Pahlavi refers to the "Pahlavi Psalter", a translation of a Syriac book of psalms from the 6th or 7th century AD. There are also inscriptions in Psalter Pahalvi in a bronze cross found in what is now Afganistan. This version of the script has 18 letters which are joined together.
Book Pahlavi is a version of the script with 12 or 13 letters, which are joined in writing, and some form complex ligatures. This is the version of the script for which there is most written material, and it was in common use until about 900 AD, after which its use was confined to Zoroastrian priests.
Beginning of the Husrō ī kavādān ud rēdak-ē:
The Parthian script developed from the Aramaic script around the 2nd century BC and was used during the Parthian and early Sassanian periods of the Persian empire. The latest known inscription dates from 292 AD.
Information about Middle Persian scripts and language
Parthian and Middle Persian written literature
Circle of Ancient Iranian Studies at the School of Oriental and Asian Studies (SOAS), University of London: http://www.cais-soas.com