Eric Bledsoe had 30 points, 11 rebounds and nine assists to lead Phoenix (47-32), which fell into a tie with Memphis for the West's eighth seed. Gerald Green scored 27 points, Markieff Morris added 20 points and Channing Frye had 13.
The Spurs outscored the Suns 37-19 in the third quarter to overcome a 21-point deficit in the first half.
Phoenix regained the lead in the fourth, taking a 98-97 advantage on Frye's 3 with 4:25 remaining but Leonard and Marco Belinelli followed with 3s to give San Antonio a lead it didn't relinquished.
After a sluggish start, Parker took over offensively to spark a 29-14 run to close the third quarter.
Parker hit a 15-foot jumper, a 3-foot driving jumper and an 18-footer before Green hit a 3-pointer to pull San Antonio within 72-68 with 6:45 left in third.
NOTES: NBA commissioner Adam Silver attended the game after sitting in on the Spurs' quarterly meeting. It was Silver's first game in San Antonio since taking over as commissioner Feb. 1 when David Stern retired. Suns G Goran Dragic (ankle) missed the game, but Phoenix coach Jeff Hornacek expects him to play Saturday in Dallas. . Phoenix scored 37 points in the first quarter, coming within 3 points of matching the most points San Antonio has allowed in the quarter set by Houston on Dec. 25.