Parker, still looking for an opponent for a mid-year fight after pulling out of a possible clash against Dereck Chisora due to the non-arrival of a contract from the United Kingdom, responded to Fury's sentiments by telling the Herald: "I heard he said something like he doesn't want to fight me because I'm like a brother to him.
"My comeback would be 'yeah, but brothers always fight and have arguments but then they peace it out and share the love again'."
Fury's relationship with heavyweight rival Parker, who successfully fought for Fury's vacant WBO world championship title in 2016, is unique.
They have been friendly for years and their closeness not only survived Parker's victory over Fury's cousin Hughie in Manchester two years ago but it was in fact strengthened by it; afterwards Tyson invited Parker out for drinks despite Hughie and his team crying foul at the majority decision in the Kiwi's favour.
Parker's promoter David Higgins is also friendly with the 30-year-old giant who is undefeated after 28 professional fights.
Parker has often spoken of Fury's generosity of spirit away from boxing and it's that which may behind the self-styled Gypsy King's reluctance to engage in combat. There are few better in the game at getting into the head of his opponent and Fury's loyalty to Parker would make such verbal attacks – never mind the physical – inappropriate.
Parker said Higgins was still working on options for his next bout, his 28th as a professional, including the potential for it to be in New Zealand next, but a fight against Chisora in London in July is probably more likely.
"It's still a possibility if everything aligns," Parker said of fighting at home again after bouncing back from back-to-back losses in the UK with a comprehensive win over Alexander Flores in Christchurch in December. "It's about getting the right people and sponsors on board and if we can pay him more than he'll get in the UK then we might switch his mind."
Asked if he was referring to Chisora, Parker said: "Chisora or anyone in the top 10. We had a good victory to come back with but we need to be fighting people who can challenge us more. We can rebuild and keep having fights that are not as hard but what's the point because you don't learn much from them."
Parker, meanwhile, laughed off accusations of cowardice from Chisora after pulling out of the proposed April fight against the Londoner.
"Parker b****ed out," Chisora said. "He must have taken too many laxatives and crapped himself. He wasn't ready for the battle and reckons he needs more time."
Asked for a response, Parker said: "I think that's the only line he has. I think someone who's been in the game for so many years should have better insults than that. I think he's losing it – he's become soft."