Odyssey Sims made a hook shot in the lane to extend Los Angeles lead to eight points with 41.1 seconds to go and Essence Carson had a block at the other end to seal it.
Chelsea Gray added 10 points with seven assists for Los Angeles (6-2). The Sparks won four times during their five-game homestand.
Parker scored 13 points in the first half to help Los Angeles to a 43-32 lead.
Tiffany Hayes scored 16 points for Atlanta (5-4), which had its three-game winning streak snapped. Jessica Breland added 14 points and Angel McCoughtry was held to 12 points on 3-of-14 shooting. The Dream shot just 38.5 percent from the floor, including 2 of 12 from 3-point range.