Parnell was born into a theatrical family in London. His father Russ Carr was a music hall artist, before becoming a theatrical agent. His uncle Val Parnell was general manager of the London Palladium.
Parnell was educated at Brighton and Hove Grammar School and studied piano from the age of five and drums for a year with Max Abrams. He made his debut, playing the drums for a concert party on the front at Scarborough in 1939. He then worked in a ballroom in Cambridge, before serving with the RAF for three years. He played in a five-piece line-up led by saxophonist Buddy Featherstonhaugh at RAF Bomber Command headquarters in High Wycombe. While he was in the RAF, he met Vic Lewis and together they formed the "Vic Lewis/Jack Parnell Jazzmen". He worked with the Ted Heath Band (1944-1951). He formed his own 12-piece line-up in 1951 and a 16-piece line-up in 1955.
During the 1940s and 1950s, he was voted best drummer in the Melody Maker poll for seven years in succession. He composed many television themes, including The Golden Shot, Family Fortunes and Love Story (for which he won the Harriet Cohen Award). In 1973, he became the first British musician to win an Academy Award, for the Barbra Streisand television special produced for ATV.
In the 1970s he formed the group The Best of British Jazz with Don Lusher, Kenny Baker, Tony Lee, Betty Smith and Tony Archer, which performed until 1985. The group had an incarnation from 1985 to December 1999, releasing two CDs: The Best of British Vol 1 (CDSIV 6146) and The Best of British Jazz Live (MER 99100CD).
Jack Parnell has three sons (including Ric Parnell who, among many other credits, played drummer Mick Shrimpton in the movie "This Is Spinal Tap") and two daughters and in 1983 moved to Southwold, Suffolk. He likes the music of Bach and playing golf, and in recent years played as part of a Jazz trio at the Green Man pub in Rackheath, Norwich, though he no longer plays drums.