Parrish attended Dundalk High School, where he was a two-way player, earning All-county honors at tight end as a senior.
He accepted a football scholarship from Temple University and played under future NFL head coach Bruce Arians .
[3] Parrish was signed as an undrafted free agent by the Miami Dolphins after the 1991 NFL draft on April 30, with the intention of playing him at guard.
On December 22, he was promoted to the active roster for depth purposes, after Mark Stepnoski was lost for the season with a right knee injury.
On October 25, 1994, he was signed by the Dallas Cowboys, to provide depth after offensive tackle Erik Williams suffered season-ending injuries in a car accident.
[18] On September 24, against the Minnesota Vikings, he had his only career start, while playing left tackle in place of an injured John Jackson.
[20] On August 29, 1996, he signed with the New York Jets to provide depth at tackle after Jumbo Elliott and David Williams were injured.
On April 1, 1998, the Montreal Alouettes of the Canadian Football League traded linebacker Henry Newby and defensive tackle Juan Hammonds to the Hamilton Tiger-Cats for the CFL's negotiation-list rights to Parrish.