The 22-year-old Fury – who has an unblemished record of 20 wins including 10 by knockout – had been due to fight title-holder Parker, also unbeaten, in Auckland on May 6th.
However, Parker's promotions company released a statement saying the former junior world super-heavyweight champion would not be able to fulfil his duties.
"Duco Events was today notified by the World Boxing Organisation that mandatory No 1 challenger Hughie Fury has an injury, and will be unable to challenge champion Joseph Parker for the Heavyweight Title on May 6 in Auckland," the statement read.
"The World Boxing Organisation has, therefore, advised Duco Events to investigate the possibility of Joseph Parker making a voluntary defence of his heavyweight title against an opponent ranked in the top 15 of the world rankings.
"Duco Events is currently investigating all options and will make no further comment today."