Joshua successfully defended his IBF title, while also claiming the WBA 'super' belt at a sold out Wembley courtesy of a dramatic 11th-round stoppage, after the Brit had been flattened in the sixth.
Sky Sports reports Parker had been due to defend his own title against Hughie Fury in Auckland on May 6, but the unbeaten Brit was forced to pull out due to a lower back injury.
The 25-year-old Kiwi will now face Romanian Razvan Cojanu on the same date, but admits after that clash he will look to arrange something with Joshua.