Palmer attended Churchill High School in Potomac, Maryland. He went on to star at Temple University.
Palmer was the runner-up to Vinny Testaverde for the 1986 Heisman Trophy and holds rushing records during his college career at Temple University. After being found in violation of NCAA policies for signing with an agent before his eligibility expired, sixteen of his marks and the six victories the school registered in the 1986 season were eliminated from the record books. He still holds several school records including career rushing yards: 3,029 (1983–85).
Palmer was selected in the first round (19th overall) of the 1987 NFL Draft by the Kansas City Chiefs, where he led the AFC in kickoff return average (24.3 yards per attempt). After having clashes with the coaching staff and not being able to take over the starting running back position from Christian Okoye, he was waived on September 4, 1989. He rushed for 607 yards in two seasons.
On September 5, 1989, Palmer was claimed off waivers by the Detroit Lions. After five games without a single rushing attempt, he was traded to the Dallas Cowboys on October 16, in exchange for an eighth round draft choice (#194-Willie Green).
In 1989, he became the starting running back after Herschel Walker was traded to the Minnesota Vikings. Palmer led the Cowboys with 446 yards on 112 carries during 9 games, on a team that would finish with a 1–15 record for the 1989 season, and that had the second lowest 16-game rushing total in franchise history (after the 2012 season), with quarterback Troy Aikman as the team's second-leading rusher with 302 yards. Palmer played a key role in the only win the team had that season (against the Washington Redskins), when he registered the only 100-yard rushing game of his career, after gaining 110 yards and scoring a touchdown.
In 1990, the Cincinnati Bengals signed him in Plan B free agency. Even though he led the team in kickoff return yardage during the preseason, he didn't make the team.
In 1991, he was drafted by the Barcelona Dragons of the World League of American Football. During the first 4 games he was the league's second leading rusher, but a hamstring injury limited him to just 39 yards the rest of the season. He finished as the team's second leading-rusher, recording 358 yards and 3 touchdowns.
On June 27, 1991, he was signed as a free agent by the Philadelphia Eagles. He was released on August 5.
Palmer finished his career professional in the World League of American Football with the Barcelona Dragons in 1992.