Tim Duncan scored 20 points and grabbed eight rebounds while Parker added 14 points and 11 assists to spark the triumph.
The Spurs became the first 2-0 team in the NBA by spoiling the Thunder's season opener in a rematch of last year's NBA Western Conference finals. The Thunder ousted the Spurs but lost to Miami in the NBA Finals.
Reigning NBA scoring champion Kevin Durant netted a game-high 23 points and pulled down a game-high 14 rebounds while Russell Westbrook, who joined Durant on the Olympic gold medal US team in August, added 18 points for Oklahoma City.
Parker sank a 3-pointer from the left corner with 28 seconds to play to lift the Spurs into an 84-84 deadlock and San Antonio took the ball after a turnover by Westbrook on the final Oklahoma City possession.
Parker then eluded Westbrook off a screen and took the ball on the left side and sank the winning shot.
Meanwhile the Brooklyn Nets will have to wait almost a month to entertain their uptown NBA rivals, the New York Knicks, due to the damage caused by Superstorm Sandy.
The league announced that the Nets-Knicks game, originally scheduled for Thursday, had been rescheduled for November 26 at the new $1 billion Barclays Center.
It will become the third game in four days for the Nets, all of them at home, with the Los Angeles Clippers visiting on November 23 and Portland following two days later, then the Knicks the next night.
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