I'm not sure about the Joshua thing. I don't know if the Furys highlighted AJ's drug conviction until the media (and E.Hearn probably) were already quite heavily implying that AJ was a "squeaky clean" saviour and that Tyson Fury was "bad" for boxing and just a bad person generally. I see no reason why New Zealand would go out of their way to accomodate Peter Fury to help Hughie Fury in an effort to defeat their sporting hero. There's no suggestion that Hughie will be denied entry, and it is Hughie who is under contract with Parker's promoters to fight Parker. Who is has in his corner is his business. He's been given the mandatory shot and he can take it or leave it. It would be very charitable for the country to make an exception. Possibly the May 6th date gives them the option to find someone else.
I think the entire world knows that the Furys are more than happy to pick up their ball and go home now. They were willing to call the Wlad fight off over a bouncy ring and a pair of ill fitting gloves. When it comes to brinkmanship, they don't really have an equal in fight game. They'll be more than happy to not take this fight and blame it on visa issues IMO. I also think Parker's team see Hughie as an easy way to get a 'big' name on his resume via his association with the lineal champ and build for big UK fights so they'll lean quite heavily on authorities who want Parker to be an international ambassador.