It was the 900th career victory for San Antonio Coach Gregg Popovich.
Parker hit two free throws with 10 seconds left in the fourth period to give the Spurs a three-point lead after they had trailed for much of the contest. Mo Williams, however, made a 3-pointer at the buzzer for the Jazz to bring about the extra period.
A 3-point shot by Manu Ginobili put the Spurs ahead for good in overtime with 2:58 left and San Antonio went on to win its fourth in a row.
Parker had missed the previous eight games with a sprained ankle. Tim Duncan added 19 points and 16 rebounds for the Spurs.
Popovich is the 12th coach in NBA history to reach 900 career victories.
Williams led Utah with 23 points. The Jazz have lost eight of 10.
Notable deaths of 2014 [PHOTOS]