Parker and Tim Duncan combined to score 41 points and rallied the Spurs from a seven-point deficit in the third quarter to steal home-court advantage in the championship series.
Miami will host Game 2 on Sunday.
Parker scored 21 points and his long jumper with 3:30 remaining gave San Antonio its largest lead of the contest at 85-79.
That set up the signature play of the game, one in which Parker -- with the shot clock running down -- stumbled and went to one knee while continuing to dribble the ball.
Parker got back to his feet and, with James contesting the shot, threw up a leaning jumper from 15 feet that went in.
Game officials looked at a video of the shot before deciding the ball had left Parker's hand an instant before the 24-second clock expired. San Antonio thus went in front by four points and the Spurs' victory was assured when Miami's Dwyane Wade missed a layup with 2.8 seconds to go.
San Antonio overcame a triple-double from James, who scored 18 points, grabbed a massive 18 rebounds and handed out 10 assists. Wade added 17 points and both Allen and Chris Bosh had 13.
Miami shot 50 percent from the field in the first half, but held only a three-point lead. The Heat wound up hitting 43 percent for the contest and forced only four San Antonio turnovers.
The Spurs were playing for the first time in 10 days after finishing off a four-game sweep of Memphis in the Western Conference finals. Miami, meanwhile, had to go the full seven games to eliminate Indiana in the finals of the Eastern Conference.