Also up in the air is the immigration status of Peter, Hughie's father and trainer and a key part of Hughie's support crew.
A serious conviction for drugs many years ago, for which Peter went to jail, could yet prevent him from getting immigration clearance, and in finally announcing the fight details today after winning the purse bid early last month, Parker's promoter David Higgins said he would support Peter's entry.
"I've had a look at it and it was some time ago - decades ago," Higgins said. "There was nothing that was in my view heinous. Yes, there were some criminal convictions for sure but I think since then... he seems to have devoted himself to training heavyweight champion boxers, and so I'm relaxed about it."
Higgins wouldn't be drawn on whether he was officially supporting Peter Fury's immigration application. "That's an issue for their camp that they're working through. It's not for us to comment."
Now that the details have been confirmed, Duco Events can go about promoting an event which will cost far more - by seven figures - than Parker's world title challenge against Andy Ruiz Jr at the same event late last year.
Several hurdles have been scaled, but more remain. A major one is that the negotiations with Fury's promoter Frank Warren, have finally been concluded, and Parker's trainer Kevin Barry today made a note of the Englishman's "stalling".
And a potential hearing into performance-enhancing drugs use by both Hughie and Tyson - the pair maintain their innocence - will not be held before the fight. Higgins also confirmed Hughie and Parker had agreed to abide by Wada's drug testing regulations for the fight.
Funding remains an issue, however. Higgins said Auckland's funding arm Ateed was not approached because he was told it would be pointless, but he was hopeful central government could come to the party for a fight which will make a big noise in the United Kingdom where the heavyweight scene is enjoying a huge upsurge in popularity.
Should the New Zealand government not commit, then Samoa, who supported Parker's win against Ruiz, will again be approached.
Parker's trainer Barry said his man returned to Las Vegas five weeks ago fit and refreshed and would next week begin sparring for the remaining eight weeks of the camp.
Barry added: "The most exciting thing for us is that after this long, stalling, drawn-out process by Fury's promoter, finally we can announce a date and start focusing on having Joe and Fury in the ring on May 6."