Parker (5-3) yielded six hits, walked three and struck out three. It was the 23-year-old right-hander's fourth consecutive appearance in which he has surrendered no more than one run.
Brandon Moss highlighted a four-run second inning with a three-run homer and former Red Sox outfielder Josh Reddick also went deep for the A's, who won the first of a three-game series.
Red Sox starter Daisuke Matsuzaka (0-3) lasted only one full inning, surrendering five runs on four hits and a pair of walks.
Dustin Pedroia knocked in Boston's only run.