Reynolds' blast in the second inning put the D-backs up 2-1 and Parra's run-scoring triple highlighted a four-run rally in the sixth, enabling Arizona to break a franchise-record, 14-game road losing streak.
Miguel Montero collected a pair of hits and an RBI, and Chris Snyder drove in a run for the victors, who had to weather a rocky ninth inning in which Aaron Heileman notched his first save despite allowing a run.
Former Tiger Edwin Jackson (4-6) collected the win, holding his ex-teammates to three earned runs on nine hits over 7 2/3 innings.
Rick Porcello (4-7) absorbed the loss for Detroit, which saw its nine-game home winning streak snapped.
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