It took just two minutes for the champions to break the deadlock as Juan Cuadrado's found Mandzukic, who saw his header saved before slotting home the rebound.
Parma should have been level in the ninth minute when Gervinho played Antonio Di Gaudio through on goal, but the striker failed to score with the net gaping.
Gervinho then restored parity with 33 minutes played, with the Ivorian turning home Roberto Inglese's cross in the box for his first goal of the season.
Parma nearly took the lead shortly before the break as Gervinho turned provider to pick out Emiliano Rigoni, whose low strike was palmed away by Wojciech Szczesny.
The second goal arrived for Juve just before the hour mark as Blaise Matuidi netted his first of the campaign with a smashing finish.
Parma looked to get back into the game and they carved out an opening in the 75th minute but Di Gaudio failed to find the target from close range.
Paulo Dybala was eventually introduced with seven minutes remaining in the encounter after surprisingly being left on the bench.
Dybala's introduction had no impact on the clash as the score remained 2-1 to the Serie A champions.