San Antonio overcame a solid professional debut from the Hornets' Anthony Davis, this year's No. 1 selection in the NBA draft.
Davis led New Orleans with 21 points and seven rebounds, helping the Hornets to a seven-point halftime lead.
The game was close throughout the second half, however, and the lead changed hands four times in the fourth period.
San Antonio seized a 94-93 advantage when Tim Duncan tipped in his own missed rebound with 1:20 to play, but Davis restored a 1-point New Orleans advantage by hitting two free throws with 1:08 remaining. They were the last points the Hornets scored.
Parker's 3-pointer moved the Spurs back in front by two and Duncan added two free throws with 24 seconds to go to account for the final score.
Duncan, beginning his 16th season with the Spurs, scored 24 points with 11 rebounds and Parker had 23 points.
Davis made 6-of-12 shots in his first game and hit all nine of his free-throw attempts. Al-Farouq Aminu added 17 for the Hornets.