Neither team scored during the final 2:23 with Golden State missing out on three chances to tie the contest. Andre Iguodala made the Warriors' final, frustrating attempt, but saw his leaning layup try rim out at the buzzer.
San Antonio shot only 39 percent and Golden State hit 41 per cent of its shots in the low-scoring contest.
Parker scored 18 points and Kawhi Leonard had 13. A 3-pointer by Patty Mills with 8:23 left in the fourth period were the last points scored by a San Antonio player other than Parker.