[1] Making his debut in 1974 at the age of 20, Parrish became a solid hitter as a third baseman in the 1970s and 1980s, hitting 20 or more home runs in a season 5 times during his Major League career.

Parrish is the only Montreal Expos player to ever hit three home runs in one game on three separate occasions (May 29, 1977, July 30, 1978, and April 25, 1980).

Parrish was traded along with Dave Hostetler from the Expos to the Texas Rangers for Al Oliver on March 31, 1982.

[2] In 1982, during his first season with the Rangers, Parrish tied a major league record held by Jim Northrup when he hit three grand slams in the span of one week.

He closed out his American Major League career by joining the Boston Red Sox during the stretch run of the 1988 season; while he was with the Red Sox, the team won the American League Eastern Division championship.

In 1996, he returned to the dugout as the manager of the Double-A Jacksonville Suns, where he won another League Championship the same year.

Parrish remained in the Tigers organization as a scout from 2000 to 2002, and in 2003, he returned to the dugout once again as the manager for the Triple-A Toledo Mud Hens of the International League.

In 2005, Parrish managed the Mud Hens to an impressive 89–55 record and the Governors' Cup as champions of the International League.