Higgins and Joshua's promoter Eddie Hearn have been in discussions for Joshua and Parker to meet in England in the spring, with Joshua holding the IBF and WBA belts and Parker holding the WBO crown.
Hearn has criticised Higgins for holding a press conference in New Zealand on Wednesday, placing emphasis on Joshua's chin and previous negotiations for the bout, but Higgins has hit back at the Matchroom Sport representative.
Higgins told Sky Sports News: "If you actually watched the press conference, what we said was correct, serious, and pure fact. What Eddie should be worried about is Joshua's chin, not our press conference. It's pathetic.
"Parker is unbeaten, he's a world champion, and he deserves 35 per cent. They will pay us 35% and not 1% less, or we'll take another fight, and we might lose, and they might fight [WBC champion Deontay] Wilder. So be the case.
"But I tell you this, Joseph Parker, his family, the trainer and me, we are united and we are adamant that's the deal. Let's make the fight."
Higgins and Parker have previously revealed that they will face Australia's Lucas Browne should they be unable to agree a deal with Joshua and Hearn.