The Heat blocked shots, made stops, and occasionally flopped, playing with renewed aggression after what coach Erik Spoelstra called a "miserable" day of watching and analyzing their passive performance from Tuesday.
They still haven't lost two in a row since Jan. 8 and 10.
Parker played through a strained right hamstring, shooting 7 of 16, but the Spurs couldn't match the Heat's speed.
After the teams traded blowouts in the previous two games, momentum swung wildly in a first half that ended tied at 49.
San Antonio raced to a quick 10-point lead, fell behind by 10 with 7 minutes left in the half, then finished with an 11-2 spurt sparked by reserve Boris Diaw. Bosh drove for a dunk that came just after the buzzer, Spurs owner Peter Holt waving it off from his seat along the sideline.
James rocked back and forth during the national anthem, a bundle of energy ready to get going.
It took a few minutes after the game started, but he began playing with the speed and power that can make him unguardable, grabbing rebounds on defense and rushing the ball up the floor himself to get the Heat into their offense.
He and Wade combined to make 10 of 11 shots and score 21 points in the first quarter, helping the Heat erase their early 10-point deficit to go ahead 29-26.
Popovich even lit into Duncan during an early second-quarter timeout with Miami on its way to a 41-31 advantage, but the Spurs had it back to even by the time the teams headed to the locker room.
Notes: Sebastien De La Cruz, an 11-year-old mariachi singer, sang the national anthem again after his Game 3 performance set off a barrage of racist tweets by what Popovich called "idiots." Popovich and Spoelstra congratulated him at midcourt after his performance, which earned him a rousing ovation. ... James passed Hakeem Olajuwon (3,755 points) and John Havlicek (3,776) to move into ninth place in career playoff scoring. James has 3,777.