The Sparks overcame a 13-point deficit. Ana Dabovic scored five points and Lavender added four during a 9-0 run that gave Los Angeles (5-14) a 78-71 lead with 3:02 left.
Jayne Appel's driving layup with 40 seconds remaining pulled San Antonio (6-14) to 80-78 but, after getting a stop on other end, the Stars came up short when Parker blocked Jia Perkins' shot in the final seconds.
Parker finished with 17 points, 11 rebounds, five assists and four blocks. Alana Beard added 15 points and seven assists, Dabovic scored 13 points and Nneka Ogwumike had 10 for Los Angeles, which is 2-1 with Parker in the lineup and now sits a half-game out of the playoff race.
Perkins led San Antonio with 24 points.