The claim, which has since been withdrawn by Parker and his promoter David Higgins, was rejected by Joshua at the London press conference staged on Tuesday to officially announce the fight.
Anthony Joshua pictured during the press conference with Joseph Parker ahead of their world heavyweight title clash.
The 28-year-old said: "It will take more than any human to stop me on this journey".
"I might give him a text and say: 'Bro do you want to walk in with me?'", Parker told the Daily Mail.
To be accused of not being able to take a punch is one of the most serious accusations in the sport and Joshua was not going to take the slur lying down.
"Now it's about making history".
Boxing trainer Kevin Barry has challenged Joseph Parker to finally produce "a career-defining performance".
"In terms of using it as a PR stunt, the rumours you have heard, they're fake news", said a fired-up Joshua.
Higgins said he had "killed a few birds with one stone" by promoting the fight and Parker by pointing out Joshua's frailties.
Joshua said, "I took my time last time out (against Carlos Takam) because I wanted to set up a good 2018".
'Anthony Joshua is a big, strong, powerful guy. If I haven't been clean for all my other fights, you'll see me struggle against Parker because anything I have been taking will be out of my system. I was very unfit but I got stopped, not dropped.
Showtime is linked to unbeaten World Boxing Council heavyweight champion Deontay Wilder, who will defend his belt March 3 at Brooklyn's Barclays Center against Cuba's Luis Ortiz. "There have been three times I have been hurt or dropped". I know he's watched me.
Watch Iain Lee talk about what really went down on I'm A Celebrity, the ups and downs of his career, and his challenges with mental health only on Unfiltered with James O'Brien. I know his strengths, he knows mine.
"I think the trash talking should still hold a sense of respect".
Parker has been on good terms with Fury since defeating his cousin, Hughie, in a defence of the WBO heavyweight title a year ago.
That was the first thing that struck me, when David Higgins was put on the spot about the "glass chin" stuff again.
Anthony Joshua himself explained to the Parker camp why he is mentally strong which is opposite from their opinion along with the questions regarding his chin. "Something that shocked us is apparently in English boxing it's not customary to do that".
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