Parker's promotional company Duco Events held a news conference on Tuesday in Auckland, where they played a compilation of Joshua being sent to the canvas.
Duco founder and director David Higgins demanded Parker should have no less than a 35 per cent split of the purse from a fight with Joshua, with March having been mentioned as a potential date for a meeting between the two.
Higgins, WBO champion Parker and his trainer Kevin Barry offered explanations why he would defeat WBA and IBF champion Joshua, who they dubbed 'Glass Jaw Joshua'.
Joshua has now offered a retort, speaking courtesy of Jumeirah Hotels & Resorts/Sports Tonight, said: "The fight's still not made… He's investing his time in the wrong areas – does he want publicity? Is it a PR stunt?
"It's not about worrying about who we fought or who we fight because I've worked my way through the ranks – from my 15th fight I competed for the heavyweight championship of the world – 15 fights in, three years as a professional.
"I think in my 19th or 20th fight we unified the heavyweight division fighting Wladimir Klitschko which in hindsight realistically… I should never have fought for the heavyweight championship but the opportunity came. And I should never have been fighting Wladimir Klitschko so early but the opportunity came.
"We managed to work with the champions many times. I think I've defended it now four or five times, with no complaints. Against one of the all-time greats we came together and put on a great show – so what's so difficult with fighting this little kid, doing press conferences talking a load of rubbish. I don't get it; we work with people. It's just very unprofessional of them.
"But it's a shame because boxing has so much credibility. The heavyweight championship is a really well respected title to have. But stuff like that takes away from the creme de le creme of the sport."
Photo by: Getty Images. Caption: Anthony Joshua took aim at Joseph Parker after a media conference held by the New Zealander