Joshua won Fury's vacant IBF title on Sunday with a knockout of American southpaw Charles Martin in the second round in London.
Listen: Martin Devlin interviews Kevin Mitchell
Should Kiwi Parker beat Carlos Takam at Manukau's Vodafone Events Centre on May 21, he becomes the mandatory challenger for Joshua's title.
However Mitchell said a looming bout between Joshua and WBA champion Fury around the middle of next year means Parker will have to wait in line to get a title shot.
Mitchell predicts Joshua will have a few easy defences before taking on Fury, meaning he won't want to fight Parker before a massive pay day.
"It would be the biggest fight in the history of British boxing. No doubt about it. Two world heavyweight champions. Presuming Fury is going to hang onto his title in the rematch if it happens against Wladimir Klischko.
"Wembley Stadium is probably not even big enough. You can imagine the hype."
Mitchell predicts the two would fight in the British summer of 2017 and says Joshua would have the edge.
"I've seen nothing really in Joshua's career to suggest that he's not going to be a terrific champion at some level. If you pushed me on it right now...it's a real close fight between him and Tyson Fury."
"I would suspect Joshua would stop him somewhere in the later rounds."