Second-half goals from Stephan Lichtsteiner and Andrea Pirlo were enough for Juve in a match that could have been postponed due to a downpour in the lead up.
Conte lost his appeal earlier in the week against a 10-month ban for failing to report incidents of match fixing when he was coach of Serie B side Siena and Juve fans made their feelings about the punishment clear with a series of chants.
Juve were awarded a penalty in the 33rd minute for a foul on an offside Lichtsteiner by Parma goalkeeper Antonio Mirante but he pushed Arturo Vidal's spot-kick aside.
Mirko Vucinic missed a great chance to give Juve the lead seven minutes into the second half when he shot straight at Mirante from a tight angle instead of rolling the ball across to Claudio Marchisio, who was standing in front of an open goal.
However, Lichtsteiner saved his team-mate's blushes two minutes later when he tapped in an easy goal from signing Kwadwo Asamoah's fizzing cross, and Juve quickly doubled their lead when Pirlo's tame free-kick was mishandled by Mirante and judged to have crossed the line.
The Parma keeper protested he had kept the ball on the line but the officials gave the goal.
"It's clear that when you concede a second goal only four minutes after the first then it's difficult," said Parma boss Roberto Donadoni.
"I don't talk about referees, they do well, do badly and make mistakes... as you saw with the penalty decision there can be mistakes.
"It something that happened and that we can't change," he added.
Two goals from Stevan Jovetic gave Vincenzo Montella a winning start as Fiorentina boss, as his team came from behind to beat Udinese 2-1 at the death.
The Tuscan outfit dominated possession, but a 28th-minute strike from the impressive Reginaldo Maicosuel gave an Udinese side missing Antonio Di Natale an undeserved lead.
Fiorentina equalised after 67 minutes when Jovetic's deflected shot looped over the Udinese goalkeeper Zeljko Brkic.
Montella's team continued to attack and got their reward when Serb Jovetic curled home a superb injury-time winner with his right foot.
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