Joshua, who became the first Briton to hold three recognized world heavyweight titles since Lennox Lewis in 1999, afterward called on Wilder to face him.
Parker's trainer backed up his fighter's claim that the referee's ability to understand English was not up to scratch.
A brass band for Anthony Joshua and a sipi tau and haka on the stage for Joseph Parker - today's weigh-in had something for everyone, including a surprisingly small 2.5kg weight advantage for the Englishman.
Matchroom view Joshua as a much bigger drawing card in the United Kingdom than Wilder is in the US. "Get him in the ring and I knock him spark out".
The Sun.co.uk said Fury released eight Instagram story videos throughout the fight in Cardiff and after Joshua's points win over Parker on Saturday night. He has already visited NY for negotiations but it gets complicated with Wilder's affairs being handled by co-managers Shelly Finkel and Al Haymon, and Haymon's business partner Luis DeCubas Jr. "I'd love to have another go; it's back to the drawing board". They can come to Cardiff, or Wembley. "I'm fit to go 12 rounds, thanks to all the New Zealanders here and back home", he added.
"The result will take care of itself, as long as I'm ready and determined and confident".
"I say my little prayer in the corner, I pray for success and I pray for his health and I pray he'll be safe once he leaves the ring", he said.
"Forget the hype, Joseph Parker is a world champion, so I knew he was going to be determined".
Hearn said: "Anthony Joshua wants all the belts - that is the big card Deontay Wilder has got to play [but] commercially Joshua is far bigger than Deontay Wilder". He is the mandatory WBO challenger and would undoubtedly be viewed as a risky opponent, but any fight would potentially lack the pulling power of a Wilder showdown. He knows he should have thrown more punches, his fitness was fine at end of fight, he wasn't breathing hard.
Huge roars ring out inside Principality Stadium as the big screens show Joshua walking into the arena. If we get two great fights, there will no doubt be a third.
In 2016, Fury vacated the WBA, WBO, IBO, and lineal titles following a medical investigation, drug issues, and two cancellations of the Klitschko rematch.
"But we were beaten by a better and bigger man".