Joshua, 28, claimed the bout in front of 78,000 people at the Principality Stadium with a unanimous points decision over the New Zealander after being taken to 12 rounds for the first time in his unbeaten 21-fight professional career.
"I want Wilder ... Get him in the ring and I'll knock him spark out Joshua said in the ring immediately after judges had scored the fight 118-110, 118-110, 119-109 in his favour.
But the London 2012 Olympic champion was in more considered mood at the post-fight press conference.
"2018 was always a time to capture all the belts," said Joshua who was flanked by promoter Eddie Hearn and trainer Rob McCracken. "We are one away now. In my eyes, it's not time to sit back and enjoy the ride."
Wilder, who is World Boxing Council heavyweight champion, took to social media after Saturday's contest between Joshua and Parker to insist he was the top boxer. “I am the baddest man in the world, the baddest man on the planet, and that's fact".
Joshua refused to respond to the challenge. "I'm not into the business of hype, hype, talk, talk," said the Briton. "I'd love to maybe go to America with Eddie and Rob and look at the landscape and see how serious they are and deal with this behind closed doors."
Hearn added: "I think a fight with Wilder has to happen in 2018. If Wilder's camp stepped up and were actually serious about the fight and serious about a deal, we are more than fair to offer them. But they are so erratic and unpredictable. I don't know what to believe."
Parker, 26, was munificent after his first defeat in 25 bouts as a professional. "We enjoyed being here. Joshua deserved to win. We're still young. We'll be back. We'll be a champion again.”