Parrott was crew chief for the No. 43 car driven by Aric Almirola for the last year. NASCAR suspended him indefinitely last week for failing a random drug test.
Car chief Greg Ebert was elevated to interim crew chief beginning with this weekend's race at Martinsville Speedway.
Parrott was the former crew chief for Hall of Fame driver Dale Jarrett. He led Jarrett to the 1999 Cup championship and two Daytona 500 wins and two Brickyard 400 wins.
Ebert has been Almirola's car chief for the past two years. He previously spent five seasons as car chief for Matt Kenseth at Roush Fenway Racing.