Trailing 2-0 in the fifth inning, the Diamondbacks got Miguel Montero and Geoff Blum aboard with none out against Chad Billingsley (4-9). Parra followed by slicing a 2-2 pitch into the left-field corner, scoring both runners.
He went to third on an out and came home with the decisive score on Stephen Drew's sacrifice fly as the D-backs grabbed a 2-1 lead in their four-game series against the National League West-leading Dodgers.
Jason Kubel added a solo homer for Arizona.
Trevor Cahill (7-7) earned the victory by limiting the Dodgers to three runs on six hits in a seven-inning performance, yielding one walk while fanning six.
Billingsley, meanwhile, lost his fifth straight start for Los Angeles, coughing up four runs on six hits over six full frames.
Jerry Hairston Jr. hit a two-run homer and A.J. Ellis went 3-for-4 with a solo shot for the Dodgers.