New Juventus coach Maurizio Sarri was missing from the sidelines as he recovers from pneumonia which will also see him miss next weekend's game against Napoli in Turin.
But defender Chiellini made the difference early, connecting with an Alex Sandro cross off a corner to fire in after 21 minutes at the Ennio Tardini stadium in Emilia Romagna.
It was just the 27th goal in Chiellini's 14-year Juventus career.
Portuguese superstar Cristiano Ronaldo drew a blank as he begins his second season in Italy with a goal ruled offside and several missed efforts.
New-look Juventus begin Serie A defence without ailing Sarri
Sarri's arrival has been hailed as the dawn of a new era for Juventus with a different exciting style of football.
But the Turin giants new-look side resembled Massimiliano Allegri's last season.
Apart from returning Argentine forward Gonzalo Higuain, back from a loan to Chelsea, preferred up front alongside Ronaldo in place of Paulo Dybala, the time was the same as last season.
Juventus's new Dutch defender Matthijs De Ligt remained on the bench with Welsh international Aaron Ramsey not in the group, while French midfielder Adrien Rabiot came on after an hour in place of Sami Khedira.
"What will Sarri say to us? He will congratulate us," said Brazilian winger Costa
"We took three important points, playing in Parma is always very difficult for Juventus.
"It was a difficult game, physical. We need to improve at the level of fitness, but we are on the right track."
Parma finished 14th on their return to Serie A last season having been demoted following bankruptcy.
Juventus coach Sarri has pneumonia, club says
But they pushed the champions hard with Robert Inglese denied twice in the first half by Juventus goalkeeper Wojciech Szczesny.
Ronaldo squandered a golden chance after half an hour controlling a Higuain cross with his chest but sending wide across the face of goal.
The Portuguese looked to have made up for his earlier error on 34 minutes, finishing off a Douglas Costa effort into the back of goal.
The goal was ruled offside after consultation with VAR with Ronaldo deemed to be marginally offside.
The five-time Ballon d'Or winner again proved wasteful after an hour latching on to a Costa cross but sending the ball over the top of goal.
His frustration continued five minutes later with Parma goalkeeper Luigi Sepe denying the Portuguese at full stretch.
Brazilian Hernani missed a chance to snatch a point for Parma rifling a freekick just wide allowing the champions off the hook to finish with all three points.
Carlo Ancelotti's Serie A runners-up Napoli are in action later on Saturday in Fiorentina.
The southerners have not won the title since 1990 and open in Tuscany with new French signing Franck Ribery set to make his debut for the hosts.
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