Playing without power forward Serge Ibaka and given two days to adjust to dropping Game 1 by 17 points, the Thunder performed worse.
Parker and Green each had eight points in the third quarter as San Antonio outscored Oklahoma City, 33-18.
It was a shocking turn for the Thunder, who started well.
Oklahoma City opened more aggressive on both ends, especially on defense. The physicality disrupted the Spurs early and frustrated them.
Duncan was issued a technical with about five minutes left in the first quarter after complaining when referee Ed Malloy ruled he threw a hip into Durant.
It turned out the Spurs veteran had little to worry about, as the Thunder finished only four for nine at the free-throw line.
The Thunder scored five straight points to open the game, with Westbrook's three-pointer capping the early run and producing what would become their largest lead of the series at 5-0.
Oklahoma City did not go under on pick-and-rolls as they did in Game 1, which allowed Parker more scoring opportunities. He took four shots in the opening five minutes. His first two baskets came on driving layups, including running right down the middle of the lane with Thabo Sefolosha trailing by a couple of steps.
San Antonio continued to struggle with its shot, but righted itself with offensive rebounding.
San Antonio closed the first half on a 25-8 run.
Green hit consecutive three-pointers to extend the Spurs' advantage to 55-44 with a minute left in the first half. San Antonio later rebounded a missed layup by Ginobili and Parker's errant three-pointer, leading to a high-arcing three by Ginobili for a 58-44 lead. Ginobili then drew a charge on Westbrook on the ensuing possession.
Durant and Westbrook opened the game shooting nine of 26 in the first half. Durant had one field goal in the final 16 minutes of the first half.
Durant made his first two field goals of the second half, but did not have another for the rest of the half.
Oklahoma City went scoreless for two minutes midway through the third quarter as San Antonio built its lead to 76-50 on a pair of free throws by Duncan with 6:20 left in the quarter.