Parker condemned Fury, cousin of former world heavyweight champion Tyson Fury, to a first professional defeat in a scrappy fight at Manchester Arena on Saturday, which the judges scored 118-110, 118-110, 114-114.
The 25-year-old has now set his sights on a unification bout with British WBA and IBF champion Joshua, who fights Bulgaria’s Kubrat Pulev in Cardiff in October.
“I feel I have better movement,” Parker said after extending his unbeaten record to 24 fights, with 18 wins by KO.
“He’s got good power and is a good champion, but if I fight him, I can bring out the weaknesses.”
Parker’s promoter, David Higgins, said Joshua’s promoter Eddie Hearn had already made contact about a fight next year.
“He wants to unify and so do we,” said Higgins, who is eager for Parker to fight again this year.
Fury’s promoter, Mick Hennessy, was furious about the decision and compared his fighter’s display to Muhammad Ali.
He also pledged to appeal against the decision.
“It was an absolute masterclass, shades of Ali. It was beautiful boxing in the heavyweight division,” Hennessy told BBC Radio 5 Live.
“Parker wasn’t even in the fight. Any punches he was throwing were either missing or hitting elbows or hitting gloves. It was very rare he got through with a clean shot. Hughie absolutely skated that fight.
“That is probably one of, if not the worst, decisions I’ve ever seen in my life. It was disgusting, absolutely disgusting.”
Fury, 23, echoed Hennessy’s words.
“I thought I’d done enough,” said Fury, who was supported at ringside by Tyson Fury.
“There are no words. At the end of the day, they are ruining people’s dreams.
“I don’t know what the judge was seeing — he kept missing. Devastating. Judges like that shouldn’t be there.”
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