Parrish helped lead the Tigers to the 1984 World Series championship, was selected as an All-Star eight times (1980, 1982–1986, 1988, 1990), and won six Silver Slugger Awards (1980, 1982–1984, 1986, 1990) and three Gold Glove Awards (1983–1985). Over his 19 MLB seasons, he compiled a .252 batting average with 324 home runs, and 1,070 runs batted in (RBIs). At the time of his retirement, he ranked fourth in major-league history in home runs by a catcher and seventh in games played at the position.
After his playing career, Parrish worked as a catching instructor, coach, manager, and broadcaster. He was a member of the Tigers' coaching staff from 1999 to 2001 and 2003 to 2005. He was the color commentator on Detroit Tigers television broadcasts in 2002. He was also a minor-league manager of the San Antonio Missions (1998), Ogden Raptors (2006), Great Lakes Loons (2007), Erie SeaWolves (2014–2017), and West Michigan Whitecaps (2018–2019).