Wilder, who holds the WBC title, took to social media following the clash, sharing his sympathy for Parker, but also accepting the undefeated Joshua's challenge.
The inexperienced referee attracted boos from the 78,000-strong crowd at Cardiff's Principality Stadium at his first world heavyweight title fight.
Deontay Wilder has accepted Anthony Joshua's invitation to fight to become the undisputed heavyweight champion of the world.
With the fight done and dusted, Joshua admitted he doesn't bear any ill will towards Parker - and made sure to check-in with his mother Sala right after the fight.
Asked what he would have to do to beat Wilder, Joshua replied: "Get him in the ring and I'll knock him spark out".
Joshua added: "That is 21 professional fights with six world title fights in it".
After his win, Joshua told Sky Sports: "My strategy was to stick behind the jab".
And he insisted he felt no disappointment in going 12 rounds.
"Joseph did get himself into a position at different times to take that fight but he was unable to pull the trigger", says Mike Angove.
"Parker said this will be a war".
The Bomber - who will fight David Haye on May 5 - sent out a very powerful tweet early Sunday morning, reminding boxing fans of Joshua's enormous achievements and stating exactly what he is; the best thing to happen to British boxing.
"If they (Wilder's management) stepped up and were actually serious about the fight, and serious about a deal we are more than fair to offer them, it could happen next".
"I would love to maybe go to America with Eddie and Rob (Joshua's trainer Rob McCracken) and look at the landscape and deal with this behind closed doors".
Parker charged out for the sixth, chasing Joshua to the ropes, and both traded big punches as a tense duel turned into a brutal encounter.
Joshua had weighed in at 17st 4lbs (110.4kgs) on Friday, well down from a career high of 18st 2lbs (115.5) against Carlos Takam last time out, while Parker came in at 16st 12lbs (102.3).
Joshua began to measure Parker with his own ramrod jab in the second, while the Kiwi was muscled to the ropes by a crunching combination in the third.