Tim Duncan added 19 points and 11 rebounds for San Antonio (3-0). Danny Green had 21 points and Kawhi Leonard 13.
Duncan and Parker combined to score seven points in the final 4 minutes to preserve the victory and give the Spurs their best start since the 2007-2008 season.
Mo Williams scored 29 points to lead Utah (1-2), leading the team's rally in the third quarter. Paul Millsap added 17 points, Al Jefferson 16 and Gordon Hayward 15.
San Antonio squandered a 19-point lead in the third quarter when Utah went on a 24-5 run to tie the game.
Williams hit a pair of free throws to cap the run and tie the game at 79 at the close of the third quarter. Williams had 11 points during the run and Millsap eight.
The Spurs then turned to their two stars in the fourth.
Manu Ginobili made his season debut with 3:06 left in the first quarter, drawing thunderous applause from the sold out crowd. Ginobili scored his first points in typical style, driving from behind the three-point line for a scooping left-handed layup. His entry in the second half drew an equally loud response from the crowd.
The Spurs shot 74 percent from the field in racing to a 62-44 lead at the half. They made 5 of 7 3-pointers in the first half.
The Jazz next play at Memphis Monday night.