During his rise through the executive ranks — from assistant vice president, district administrator, branch administrator, senior vice president, and assistant to the president — he was exposed to all phases of UCB's business. In 1967, he was promoted to executive vice president, and in April 1968, Barker was named president. In 1971, he assumed additional responsibility as CEO, and by 1978, he was chairman and CEO.
He served as a Trustee of the University of Chicago, a member of its Citizens Board, and a member of the Council on Chicago Booth. He was chairman of the board of trustees of Occidental College in Los Angles, a trustee of the Los Angeles County Museum of Art, and an overseer of the Henry E. Huntington Library and Art Gallery, and was affiliated with a variety of organizations devoted to business, education, and public service.