The Rockets found themselves trailing 106-103 with less than a minute remaining in the game but then went on a 9-0 run to grab the lead.
Harden responded with a power move at the other end. His layup rimmed out, but Beverley crashed the boards to tip the ball in over Tim Duncan, giving the Rockets a 108-106 lead.
Parsons finished with 25 points. Dwight Howard added 13 and 11 rebounds.
Parker provided 27 points, eight assists and six rebounds for the Spurs, who dropped their first home game of the season.
Duncan scored 20 points and grabbed eight rebounds.