After Martin Prado bunted into an out and Paul Goldschmidt was intentionally walked, Hill lashed a score-tying hit into left field.
The contest remained knotted until the 14th when Parra and Prado drew consecutive walks against Bud Norris (8-10), and after Goldschmidt fanned, Hill came through again, this time grounding up a pitch up the middle for the game-winner.
Heath Bell (4-1), the last of seven Arizona pitchers, notched the decision as the D-backs swept their three-game interleague series, staging late rallies and winning in walk-off fashion in each of the victories.
Parra, Prado and Matt Davidson each drove in a run for the second-place Diamondbacks, who come into the contest trailing the National League West-leading Los Angeles Dodgers by 7 1/2 games.
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