Parker (10-8) yielded just three hits, walking two and striking out two while becoming just the 10th rookie in Oakland history to win 10 games.
Ryan Cook tossed a scoreless eighth and Grant Balfour collected his 17th save with a 1-2-3 ninth for the American League wild card-leading Athletics, who won for the 13th time in 16 contests and took the first of a key four-game series.
Brandon Moss cracked a solo homer in the fifth inning to break a 1-1 tie and Chris Pennington added another bases-empty blast in the next frame for the A's, who snapped the Angels' six-game winning streak.
Dan Haren (10-11) was the losing pitcher, allowing three runs on four hits over 6 1/3 innings.
Torii Hunter drove in the only run for Los Angeles, which fell 1 1/2 games behind idle Baltimore for the second AL wild card spot.
Celebrity Couples of 2014 [PHOTOS]