Parker went 2-of-8 from the field during the opening three periods and managed only five points before hitting the key shots down the stretch that brought the Spurs their fourth win in five games to open the season.
The Suns took a one-point lead with 1:45 to play, but Parker put the Spurs back in front with a jumper that came 17 seconds later.
A layup by Miles Plumlee again pushed Phoenix ahead and Parker responded with a jumper for a 97-96 San Antonio advantage with 37 seconds left.
Markieff Morris came off the bench to lead the Suns with 23 points and 12 rebounds.
2014: The Year in Music [PHOTOS]