Mario Mandzukic opened the scoring for the defending Serie A champions after only two minutes.
But Parma responded well and were level before the break when Gervinho marked his full debut with a close-range goal.
Cristiano Ronaldo is still awaiting his first goal since joining Juve, but his side claimed the win through Blaise Matuidi’s second-half strike.
Mandzukic quickly broke the deadlock by taking full advantage when a rebound off Simone Iacoponi fell kindly for the Croatia striker inside the six-yard box.
Parma’s Leo Stulac hit the crossbar with a brilliant free-kick before Gervinho’s equaliser, the ex-Arsenal striker diverting the ball home past former team-mate Wojciech Szczesny.
A brilliant touch from Mandzukic set up the winner, his deft backheel allowing Matuidi the chance to hit a powerful shot into the roof of the net from a tight angle.
Juve went close again when Douglas Costa hit the post late on.
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