Palmer was a lower order right-handed batsman and a right-arm medium-pace bowler. He made his first-class and List A cricket debuts as a 16-year-old for Somerset in 1982, making 27 in the second innings of his first first-class game. In 1983, he played in 10 first-class games and made both his highest score and took his career-best innings return in the season. In only his third first-class match, against Gloucestershire at Bristol, he made 78 and put on 124 for the seventh wicket with Peter Denning. Then, in the last match of the season against Warwickshire he took five wickets for 38 runs in the first innings, and this would be the only five-wicket haul of his first-class career. In 1984, with Somerset's overseas players Viv Richards and Joel Garner absent playing for the West Indies touring team, Palmer played fairly regularly for both the first-class and List A sides. Wisden noted, though, that he "did not make the progress hoped for" and that, though "a spirited competitor, he possible needs more positive thought and application to bring greater consistency". In 16 first-class games, he hit 299 runs at an average of 16.61 with only one score of more than 50 and took 30 wickets at an average of 41.03 runs per wicket. Success was even more elusive in the one-day games: in 20 matches, he took only nine wickets at an average of more than 70 runs per wicket.
Despite this recognition, Palmer's career at Somerset failed to take off. In 1985, he was restricted to eight first-class matchesand 10 List A games; in 1986, there were just four and nine of each. His best performances in these seasons came in one-day matches. Against Kent at Canterbury in the John Player League in 1985, he took five wickets for 34 runs, his first five-wicket haul in limited-overs cricket. And the following year in the Benson and Hedges Cup match against Sussex at Hove he scored 53, his only innings of more than 50 in List A cricket. The departure of major cricketers – Richards, Garner and Ian Botham – after Somerset's disastrous 1986 season gave Palmer more opportunities in 1987, but his impact was confined largely to the one-day game and in the match against Kent at Canterbury he improved on his best bowling figures from two years before, taking five for 24: these remained the best List A bowling figures of his career. But in 1988 he was dropped from the team for many games and after the 1989 season, when he played only one-day matches for the team, he left the Somerset staff. After leaving Somerset, Palmer played Minor Counties cricket for Bedfordshire and Oxfordshire, second eleven cricket for Derbyshire, minor matches for Dorset and lesser matches for Marylebone Cricket Club. But he did not reappear in either first-class or List A cricket.
