Tyson Fury even joked about his weight with a reference to 26 stone. His challenge is in jest as his mental health issues have led to his boxing licence being withdrawn. Many pundits doubt that he will ever fight again.
Wilder, the WBC champion since early 2015, says he is ready to step up as the replacement for Hughie Fury, the young Brit who pulled out of the WBO title fight in Auckland on May 6.
Wilder wants to fight Parker this year, although May is too soon for him to prepare. He wants the winner to then fight the victor of this weekend's Wembley clash between Anthony Joshua and Wladimir Klitschko.
"Forget all them other scrubs... Right now you're talking about a champion versus another champion, let's make a unification bout at the end of the year, we'll get a mega fight against the winner of Klitschko and Joshua," Wilder says in a social media post.
"Joseph Parker, if you're really serious about fighting somebody, if you're really serious about unifying the division, if you're really serious about giving the fans what they want to see, then what other person is available, than me."
Wilder also casts doubt on the integrity of the Hughie/Tyson Fury camp, saying: "I hear about this Hughie Fury backing out two weeks before a fight. Ya'll already know they've got a history of doing stuff, family history at that."
He also took a shot at Fury's reason with pulling out of the Auckland fight, a back injury which his camp claim he's struggled with the last three weeks.
"I heard about the injury. They said it was a lack of self confidence, that sounds about right - that's ridiculous."
The WBO say Parker can make a voluntary defence of his title, but must choose a credible fighter from their top 15 who is not already engaged to fight.
American Dominic Breazeale, ranked sixth, has put his hand up to replace Fury. Wilder does not figure in the WBO's top 15. The 31-year-old, from Alabama, has won 37 of his 38 fights by knockout. He won the bronze at the 2008 Olympic Games.
Watch: Parker on Fury pulling put of title fight