Parker (4-6) registered a victory in his second straight start, surrendering just two hits and two walks over 6 1/3 innings. The right-hander struck out seven.
Jerry Blevens and Ryan Cook combined for 1 2/3 scoreless relief frames and Grant Balfour racked up his 13th save as Oakland reeled off a ninth win in 10 games, completing a sweep of its three-game series against the Chisox.
Josh Donaldson got the Athletics' only RBI with a sixth-inning sacrifice fly and Oakland added an insurance run on Chicago outfielder John Danks' fielding error in the eighth.
Chris Sale (5-3) was saddled with the loss after allowing just one run on four hits over six frames.