Parker scored the final four points of the half, the last a layup at the buzzer to push the lead back up to nine.
Green and Duncan each had 13 first-half points to lead the Spurs. San Antonio's Big Three -- Ginobili, Parker and Duncan -- combined for 35 points in the first half.
James topped the Heat with 16 first-half points, but he struggled in the second half, a big part of why the Heat face elimination heading into Game 6.
The league MVP hit eight of 22 shots, and the Heat made 43 percent overall. San Antonio wound up hitting 60 percent of its shots.
"We can't worry about a Game 7," James said. "We have a Game 6. It is what it is. We have a Game 6 on our home floor. I have to come up big for sure in Game 6. I believe we all have to play at a high level to keep this series going.
"I always just focus on the present. The next challenge is on Tuesday night. We have an opportunity."
Allen added, "Everything that we've done all year has come to this point right here. We have to win. Over the course of this year, we've found ourselves in so many different situations. We've thrived in tough moments."
NOTES: The Spurs announced that G Patty Mills, who had surgery Saturday to remove an abscess on his right foot, would miss the remainder of the Finals. The 24-year-old appeared in 22 postseason games, averaging 1.5 points and 3.3. minutes. ... Thursday's combined 85-point performance by James (33), Wade (32) and Bosh (20) was the highest single-game scoring trio in Heat postseason history. The last team trio to have at least two 30-point scorers and a 20-point scorer in the Finals was the 1995 Houston Rockets with Hakeem Olajuwon (34), Sam Cassell (31) and Clyde Drexler (23) in Game 2.