Oakland also sent lefty Ian Krol to the Nationals as the player to be named in the three-team trade that sent John Jaso to the A's, Michael Morse to the Mariners and two other Minor Leaguers to Washington.
Byrd, 26, spent three seasons in the Brewers' farm system, most recently compiling a 2.59 ERA in 50 relief appearances for Double-A Huntsville. He appeared in one game this spring before he was returned to Minor League camp on March 7.
Parker, taken by the A's in the fifth round of the 2009 First-Year Player Draft, hit .256 with seven home runs and 47 RBIs in 99 games for Triple-A Sacramento last season, after batting .287 between two affiliates the year before.
Over four minor league seasons, the 25-year-old Parker is a career .277 hitter with 43 HR and 262 RBIs in 448 games.
Jane Lee is a reporter for MLB.com. Read her blog, Major Lee-ague, and follow her on Twitter @JaneMLB. This story was not subject to the approval of Major League Baseball or its clubs.