The family said Parnell died at his home in Southwold, eastern England, on Sunday following a yearlong battle with cancer.
Parnell was born in 1923, the son of a showbiz family his father was a music hall performer and his uncle ran a string of theaters and began drumming professionally as a teenager. During World War II he served in the Royal Air Force and performed in a band at the headquarters of Bomber Command.
Later, Parnell joined the renowned Ted Heath jazz band before leading his own ensembles.
As musical director at British broadcaster ATV from the late 1950s, he oversaw the music for long-running variety show "Sunday Night at the London Palladium," produced specials featuring Tom Jones and Barbra Streisand, composed theme tunes and served as musical director of "The Benny Hill Show."
In 1976, ATV began producing "The Muppet Show," a musical variety show with a cast of Jim Henson puppets and celebrity human guest stars.
Parnell conducted the orchestra for the whole of the series' five-year run, although the ostensible bandleader was the pop-eyed Muppet conductor, Nigel.
Parnell retired from ATV in 1982 but continued to perform with bands near his home well into his 80s.
He is survived by his wife, Veronica, two daughters and three sons two of them drummers.
Funeral details were not immediately available.