Listen: Deontay Wilder talks to the Radio Sport Breakfast
The 31-year-old Wilder, speaking from Nevada, said: "Put your money where your mouth is, put your belt where your mouth is.
"I was trying to be his (Parker's) next opponent. Why would you want to fight anyone else? I'm just tired of sitting around.
"A lot of promoters and managers have got to stop being scared, everyone has got to lose some time. It's an unpredictable sport and that's what I love about it.
"I don't care where it is...here, there, on the moon. It don't matter to me. I don't care if it is one people or 100,000 watching, it is one ring, one opponent.
"I'm not scared, it's perfect. Why would you want to fight anyone else?"
Meanwhile the war of words involving the Fury camp has continued, with fallen champ Tyson Fury attacking Wilder in response to Wilder's attack on Hughie Fury.
Tyson Fury tweeted: I see old bronze balls (Wilder) is spouting s@#$ again."
Wilder has claimed a lack of confidence, rather than injury, led Hughie Fury to pull out of the Auckland title challenge.