Parker (1-0), who held the Chicago White Sox to one run over 6 1/3 frames in his first appearance April 25, stymied Boston on a single run, tossing 59 of his 98 pitches for strikes.
The 2009 first-round draft pick of the Arizona Diamondbacks allowed four hits and a pair of walks while striking out four.
Jordan Norberto got the final two outs of the game to record his first career save.
Jemile Weeks keyed a four-run fourth inning with a two-run single and then came around to score on a wild pitch.
Felix Doubront (1-1) was touched for five runs on six hits in a four-effort for the Red Sox, who lost for the second time in three games.
Boston's Mike Aviles drove in a pair of runs.
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