He studied at the Dallas Art Institute from 1955 to 1956 but didn't become interested in cartooning until he later visited two of the most accomplished editorial and sports cartoonists in Texas: Herc Ficklen and Bill McClanahan. He quickly became interested in their work and often visited them at the Dallas Morning News. They were confident in Palmer's artistic capabilities and encouraged him to pursue a career in editorial cartooning.
In 1968, with the help of Ficklen and McClanahan, Palmer landed a job in Montgomery working as the cartoonist for the Alabama Journal. He stayed there until 1972, when he replaced McClanahan at the Dallas Morning News. He shared cartooning duties with Ficklen and then later Bill DeOre.
In 1978, Palmer moved back to Alabama to serve as cartoonist for the Montgomery Advertiser. He enjoyed over 25 years of service at the Advertiser, where he retired from full-time work in 1998. He continued to work part-time at the newspaper until 2003.
He and his wife, Katie, live in Montgomery, Alabama. They have two children and two grandchildren.