The Heat led 72-69 at the end of the third quarter but the defending NBA champions were out-scored 23-16 in the fourth and Parker sealed the victory with a superb late basket as the shot clock expired.
With the Spurs possessing a two-point advantage and on their likely final play of the game, Parker dribbled and wriggled and yet was still faced with the daunting presence of LeBron James in front of him.
With time and space running out, Parker slipped to his knees but managed to leap back up and find the basket with a jump shot from 16 feet (4.8 metres) that put San Antonio four points up with just 5.2 seconds remaining.
"We were very fortunate. It looked like he had lost it two or three times... but he stuck with it," Spurs head coach Gregg Popovich said.
"He got control of it again and then got it up there on the rim. It was a great effort by Tony," he said.
Frenchman Parker had a game-high 21 points with six assists while Miami's LeBron James, the league's Most Valuable Player, had his lowest scoring game in this postseason with 18.
James had just eight points in the entire second half as the Spurs maintained their 100 per cent record in winning the opening game in five NBA Finals.
The win effectively wipes out Miami's home advantage in the best-of-seven series, which continues in South Florida on Sunday.
Tim Duncan, searching for his fifth championship title with the Spurs, finished with 20 points, 14 rebounds and four assists.
Miami led for the bulk of the game but San Antonio, who only gave up four turnovers in the entire contest, grabbed a 77-76 lead on two free throws from Parker with 7:47 remaining.
A Danny Green three-pointer with a little more than two minutes remaining extended the advantage to 88-81 but Ray Allen kept Miami in the contest, putting away all three free throws after being fouled attempting a shot from outside the arc.
After Duncan sank two more free throws, Miami's Chris Bosh crucially missed his fourth three-point attempt in as many tries.
James, however, maintained his composure to put away a pair of free throws to bring Miami within two points but then came Parker's moment of magic.
"There were plays to made and our offensive inefficiency down the stretch, most of the fourth quarter, probably hurt us more than anything," Miami coach Erik Spoelstra said.
"Now we have to gather ourselves and get ready for Game Two," he added.