He played college baseball for the Arizona State Sun Devils, then was selected by the Boston Red Sox in the ninth round of the 1976 amateur draft.

In 1978, his third season of professional baseball, he was named the International League's All-Star catcher and Most Valuable Player after slugging 20 home runs and hitting .299 in 133 games played.

During his rookie 1979 season in the Majors, Allenson was the Red Sox' most-used starting catcher, filling in for injured veteran and future Baseball Hall of Famer Carlton Fisk.

After a 31–37 year as skipper of the Bluefield Orioles in 2006, he returned to the International League to manage the Norfolk Tides to a combined 228–254 in a little over 3+1⁄2 seasons.

[8] Allenson rejoined the Toronto organization in 2013 as manager of the New Hampshire Fisher Cats, leading that edition to a 68–72 mark, then was promoted to Buffalo, where his Bisons teams went 77–66 (2014), 68–76 (2015) and 66–78 (2016).