With the score tied 1-1, Baltimore's Darren O'Day (5-3) took the mound to start the ninth and Crisp greeted him by bashing a pitch over the wall in right field, notching his third straight game with a home run.
Parker (10-6) dominated the Orioles on one run over eight innings, yielding five hits and three walks while striking out five as the A's evened their three-game series.
The victory gave Oakland a three-game lead over Baltimore in the race for the final American League wild-card spot.
Jed Lowrie accounted for the A's other run with a sixth-inning RBI double.
Orioles starter Chris Tillman settled for a no-decision after surrendering just one run on three hits over eight innings. He walked two and struck out nine.
Ryan Flaherty homered for Baltimore.