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 draft 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.