In doing so, they ended the winning streak of a Torino side hoping to push for a European place in the second half of the season.
Torino ended last year by winning three in a row to move up to seventh in Serie A, and they were off to a flyer when Antonio Mirante put them 1-0 up after 21 minutes.
But Marco Marchionni levelled before Alessandro Lucarelli put Parma ahead on the stroke of half-time, with Amauri wrapping up a 3-1 win 20 minutes from time.
Parma threatened first when Marco Parolo's cross-shot tested goalkeeper Daniele Padelli, but he was able to push it to safety and the visitors were ahead when Mirante met Matteo Darmian's cross to finish from close range.
It took until the 34th minute for Parma to level, hitting home a volley from Mattia Cassani's pass, but once they had further chances swiftly followed with Amauri striking the crossbar before Jonathan Biabiany saw a shot go narrowly over.
A second goal seemed likely, and it arrived on the stroke of half-time when Lucarelli's clever back-heel diverted Nicola Sansone's corner into the net.
Substitute Riccardo Meggiorini led Torino's search for an equaliser after the break but he headed over from a corner on the hour mark and then blazed well over with a shot from distance.
Parma punished those misses when they scored the third in the 70th minute with Amauri breaking into the area before slotting the ball into the net.
Giovanni Pasquale and Migjen Basha had further chances for the visitors but their winning run came to an end as Parma closed the gap in the table.