Parker has been with the company since June 1995, when he joined as Senior Vice President and Chief Financial Officer of America West Airlines. He additionally held titles as Executive Vice President, corporate group, and Chief Operating Officer before being elected Chief Executive Officer and Chairman of America West in September 2001.
He continued as CEO of US Airways following the merger with America West Airlines in 2005 until 2013 when he was elected CEO of American Airlines Group, Inc. and its principal subsidiary company American Airlines he is replaced by American's executive vice president and chief operating officer and US Airways chief operating officer Robert Isom. Under his tenure as CEO, he made an unsuccessful bid to merge US Airways with Delta Air Lines in 2007, and with United Airlines in 2010.
Before joining America West Airlines, Parker was at American Airlines from 1986 to 1991, where he was a member of then CEO Robert Crandall's, "Brat Pack", alongside Thomas W. Horton (CEO now Chairman of American), David Cush (CEO of Virgin America) and Ben Baldanza (CEO of Spirit Airlines), and was Vice President and Assistant Treasurer at Northwest Airlines from 1991 to 1995. On February 14, 2013, US Airways and American Airlines announced their long-anticipated merger, with Parker taking the CEO role in the combined company.