Memphis climbed within one midway through the fourth quarter before San Antonio ran off eight consecutive points, the first six coming from Parker.
Parker finished with 20 points in what was San Antonio's ninth straight victory. The Spurs kept pace with Indiana for the NBA's best record at 11-1.
Tiago Splitter had 17 points and seven rebounds for the Spurs and Manu Ginobili added 15 points.
It was the 10th win for the Spurs in their last 12 meetings with the Grizzlies.
Swim Week Miami Beach 2014 [PHOTOS]