Atlanta had a chance to tie it but Shoni Schimmel's three-pointer was short and after the buzzer.
Parker scored 26 points and Jantel Lavender added 15 as the Sparks moved closer to a playoff berth. Nneka Ogwumike had 14 points and 12 rebounds as the Sparks spoiled the homecoming of Michael Cooper, who made his first appearance at Staples Center as Atlanta coach.
Cooper, who is recovering from surgery for tongue cancer, served two stints as Sparks coach and guided them to WNBA titles in 2001-02.
Atlanta pulled to 78-77 with 4.7 seconds left on Schimmel's three-pointer before Parker made both free throws. Kristi Toliver's three-pointer from the right side gave the Sparks the lead for good at 76-74 and Atlanta was forced to foul.
Erika de Souza scored 21 points to lead the Eastern Conference-leading Dream, who lost their fourth straight game.