In linguistic classification, the language used and spoken in Iran between 3rd century B.C. and 7th century A.D. is referred to as "Middle Persian". Persian(Farsi) is the Middle era dialect which was prevalent in the central parts of Iran during Arsacid (Parthian) and Sassanid empires. Middle Persian(Farsi) grammar was simpler than Old Persian grammar, but more complex than the Post-Islamic era Persian.
The word "Pahlavi" (Pahlavi, پهلوی in Persian) means "attributed to Pahloo ((پهلو, which is a form of "Part" (pārt, پارت in Persian) and was the language of Parthian people. "Pahlavi" language or "Middle Persian" (Dari Persian) is a direct intermediary between Old Persian and Contemporary Persian, from which Modern Persian has originated.
There are lots of lexical and grammatical resemblances between modern Persian and Middle Persian. It should be noted that the linguistic evolution which lasted from the era of Old Persian to the era of Middle Persian is far greater and more influential than the one that happened to Middle Persian, resulting in the advent of Modern Persian.