As Cristiano Ronaldo was involved in a series of near misses, Chiellini netted in the 21st minute following a corner.
Chiellini and fellow center back Leonardo Bonucci were preferred by new coach Maurizio Sarri over new signing Matthijs de Ligt, even though the Bianconeri spent 75 million euros ($85 million) to acquire the Netherlands defender from Ajax last month.
Sarri was absent due to pneumonia but there were still signs of the former Napoli and Chelsea coach's quick passing game as Juventus attacked relentlessly.
Ronaldo had a goal disallowed in the first half by the VAR due to an almost imperceptible offside call.
While Ronaldo missed wide and over on several occasions in the first half, he also placed a powerful, long-range blast on goal near the hour mark that required a difficult save from Parma goalkeeper Luigi Sepe.
Juventus keeper Wojciech Szczesny also produced a tough save early on to deny Roberto Inglese after a counterattack.
As expected, former Juventus captain Gianluigi Buffon remained on the bench after returning from one season at Paris Saint-Germain.
Due to oppressive heat and humidity, cooling breaks were given midway through each half.
Later, title challenger Napoli was playing at Fiorentina. Next weekend Juventus hosts Napoli.