The Wizards visit Cleveland on Tuesday in their regular-season opener.
Washington sat its starters for much of the fourth quarter after leading most of the game.
Jackson hit a 3-pointer to cap a 9-0 run at the close of the third quarter as the Spurs took a 74-66 lead after a slow start.
Washington held a 40-27 lead with 9 minutes left in the first half following an 18-8 run. The Wizards built their lead as the Spurs' starters sat for about eight minutes bridging the first and second quarters.
San Antonio rallied once its starting lineup returned, with Parker scoring eight points in an 18-5 run.
Boris Diaw scored on a fast-break layup to give San Antonio a 61-59 lead, its first since holding a 4-2 advantage. Parker set up Diaw's basket with a diving save out of bounds on a Washington miss.
The Spurs were without Manu Ginobili, who sat out with back spasms.