Denver led by one in overtime when Houston made three consecutive 3-pointers to make it 130-122 with less than two minutes left. Parsons made the last two and high-fived with fans on the front row as he ran down the court after sinking the second one.
Houston had a franchise-record, 47-point second quarter, but Denver overcame a 17-point halftime deficit to tie it at 88 entering the fourth period.
The Rockets then trailed by eight late in regulation before using a 9-1 run, capped by a 3-pointer by Harden under heavy pressure from Randy Foye, to tie it at 117 with 7.1 seconds remaining.
Foye made one of two free throws for Denver before Harden's shot. Foye had a chance to win it in regulation, but his jump shot was short.
Evan Fournier scored 26 points and Aaron Brooks added 24 points and 15 assists against his former team. Brooks, who was traded to Denver on Feb. 20 for Jordan Hamilton, started for Ty Lawson, who was out with a left ankle sprain.
The game was still tied with about nine minutes to go when the Nuggets scored 11 straight points to make it 106-95 midway through the quarter. Fournier made two 3-pointers in that span and Brooks had one.
Jeremy Lin scored 19 points, Omer Asik had a season-high 18 points and tied a career best with 23 rebounds, and Parsons finished with 15 points.
Quincy Miller scored a career-high 19 for the Nuggets and Kenneth Faried added 16 points with 13 rebounds.
The Rockets, who have already clinched a playoff spot, were playing without Dwight Howard and Patrick Beverley, who both missed their fifth straight games. Howard has missed eight games recently because of a left ankle strain and Beverley is out indefinitely with a torn meniscus in his right knee.
The game was tied at 88 entering the fourth quarter.
Houston was ahead by 15 when Denver took over, using a 13-2 spurt to get within 79-75 with four minutes left in the third. Miller led the Nuggets in that span, scoring the first seven points of that run. Houston missed three shots and had three turnovers as Denver cut into the lead.
A 14-4 run by the Rockets to start the second quarter gave them a 36-31 lead. Omri Casspi and Isaiah Canaan both made 3-pointers in that stretch. Brooks made a layup after that before Casspi hit another 3-pointer to make it 39-33 with about eight minutes left in the first half.
The Nuggets got within a basket after a jump shot by Brooks about 2 1/2 minutes later. But Houston responded with a 9-2 run, powered by five points from Harden, to extend the lead to 52-44.
Lin had seven points in the last 42 seconds of the first half, including a 3-pointer at the buzzer, to push Houston's lead to 69-52 at halftime.
NOTES: The Nuggets made 14 3-pointers, with six from Fournier. ... Canaan had a career-best 15 points.