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Manager Kirk Gibson made the announcement before Game 1 on Saturday.
The Diamondbacks' top pitching prospect tossed 5 2/3 shutout innings in his major league debut against the Los Angeles Dodgers on Tuesday night, but did not get a decision in the Diamondbacks' 10-inning, 7-6 victory.
Parker was the Diamondbacks' first-round choice (ninth overall) in the 2007 June amateur draft. The right-hander underwent reconstructive surgery on his elbow and missed the entire 2010 season. He was 11-8 with a 3.79 ERA in 26 starts for Double-A Mobile this season.
Arizona has seven relievers on the roster, with six right-handers and left-hander Joe Paterson.
Left-handers Zach Duke, Wade Miley and Alberto Castillo, and right-hander Ryan Cook did not make the NLDS roster.