The 22-year-old cousin of former world heavyweight champion Tyson Fury was set to face WBO champion Parker in Auckland, New Zealand, on 6 May, in his first world title fight.
A statement from Team Fury and Hennessy Sports said Fury had been advised to "rest for three to four weeks".
Parker's promoters, Duco Events, said work had started to find a replacement.
The Fury statement continued: "Understandably, Hughie is devastated by the injury which has temporarily halted his dream of fighting for the world heavyweight title - but the fight will be rescheduled and a new date will be announced shortly."
Hughie Fury has pulled out of his world heavyweight title fight against Joseph Parker after suffering a back injury.