Parker fought in the UK for the first time earlier this year, beating his WBO mandatory challenger Hughie Fury on points in Manchester.
The fight was supposed to set up a unification bout with Joshua, who successfully defended his WBA and IBF straps against Carlos Takam at the end of October.
However, Parker was unimpressive against Fury and is not regarded as a huge pay-per-view pull by Joshua’s camp, meaning they will only make the fight if the Brit is given a significant majority of the purse.
But Parker’s promoter David Higgins has become increasingly frustrated by the “insultingly low” offers tabled by Matchroom Boxing promoter Eddie Hearn and has issued an apparent ultimatum.
“We are at 35 per cent and they’ve sort of come up to 30 per cent which means the ball is still in their court,” Higgins told World Boxing News.
“We’re not going to move from 35 per cent so that’s where it sits.”
Higgins, who last weekend provided evidence of what he believes to be Joshua’s weak chin, also wants a rematch clause in the contract.
“What’s been overlooked is Eddie Hearn criticising our press conference but totally avoiding the point of it, which means what we are saying is true,” Higgins added.
“Joshua has been flattened ten times and he knows it.
“Eddie and AJ have skilfully avoided even going near that topic.
“They sat down with their PR and played on the narrative that we did a crap press conference, whilst totally avoiding the two most important things, mainly that Joshua does have a dodgy chin and the other being this is a worldwide unification bout for which he gets 65-35 and a rematch clause.
“I must admit they were pretty good how they handled it, but if you have Joshua saying he wants a legacy unification fight and he turns down 65 per cent of net profit – plus a rematch clause – I think that would be embarrassing.”