Gary Neal added 20 points and Tiago Splitter finished with 16 points and eight rebounds for San Antonio (19-6), which overcame the loss of Manu Ginobili to improve to 8-2 at home. Danny Green scored 12.
Ginobili departed with a bruised left quad after he ran into Chris Wilcox's knee on a screen with 2:42 left in the first quarter. Spurs coach Gregg Popovich said he didn't have an update on the reserve guard after the game.
Spurs star Tim Duncan had a season-low five points, making Parker's effort even more important. Duncan had nine rebounds, but was 2 for 13 from the field.
Parker's layup with 3:35 remaining gave San Antonio a 96-82 lead, leading to his animated celebration and prompting a timeout by Celtics coach Doc Rivers. Both teams pulled their starters a minute later.
"That was an amazing shot," Neal said. "It looked like he lost his balance and he was able to still put the spin on it to get it to go in on that. Tony's a phenomenal player. Through the years I've been here, he's done some phenomenal things."
Paul Pierce and Jason Terry each scored 18 points for Boston (12-11). Kevin Garnett added 13 points and Courtney Lee had 11.
"(Parker) has been in those situations many times," Pierce said. "He understands when to take on the trap. He knows when to get rid of the ball. He has a great in-between game. That is what makes him one of the premiere point guards in the league."
San Antonio went on a 17-8 run after Boston closed to 79-74 on a three-point play by Lee with 9 minutes remaining.
Parker had nine points, one assist, one rebound and a steal during the run.
"Obviously, Tony was unbelievable down the stretch, but a lot of guys like Gary, Danny and Tiago played very solid basketball," Popovich said.
Splitter scored 10 points in the first half, going 4 for 5 from the field. With Duncan struggling, the Spurs needed Splitter's production.
"Tiago came in and played great," said Parker, who had one turnover in 36 minutes. "It's going to be big for us because we can't play Timmy 40 minutes every game. We have to rest him for the playoffs, so if Tiago can give us quality minutes like that, it's definitely going to be helpful."
The Spurs went on an 11-0 run after Jeff Green's layup tied it at 55 in the third. Danny Green hit a pair of 3s and Neal had a 3 and a teardrop jumper during the spurt.
The Spurs were 15 for 20 on free throws compared to just 6 for 8 for the Celtics. The discrepancy didn't go unnoticed.
"They went into the fourth quarter with six fouls total," Rivers said. "That means they are one heck of a defensive basketball team. It is what it is. They shot more free throws and two of our free throws were just technical. It's hard to win games on the road when a team goes to the line like that and we don't go to the line."
NOTES: Boston, which lost 101-89 at Houston on Friday night, will play 21 back-to-back games this season, but none consecutively at home. The Celtics had split their first six back-to-backs this season prior to Saturday's loss against the Spurs. ... San Antonio swingman Stephen Jackson (broken right pinkie) and F Kawhi Leonard (left knee) have both been out a month with injuries. ... San Antonio leads the NBA with 25.3 assists per game, its highest total since averaging 24.9 assists in 1995.