The two champions will meet on 31 March after drawn-out negotiations reached a successful conclusion
Joshua, 28, holds the IBF and WBA belts, while New Zealander Parker is the WBO champion.
Cardiff's Principality stadium - where Joshua successfully defended his titles against Carlos Takam in October - will host the bout.
"I'm delighted to get this fight made - it's been a long time coming," said Joshua's promoter Eddie Hearn.
"It's the first time in history that two reigning heavyweight world champions have met in Britain."
Joshua added: "These fights aren't easy because there is a lot on the line - so respect to team Parker for taking the challenge.
"I'm looking forward to it. Training camp is under way and before you know, 31 March will be upon us."
Parker's representative David Higgins arrived in the UK last week to finalise a deal. Talks had previously stalled over how the purse would be split but a compromise was reached last month which will see Parker, 25, earn 30-35%.
Parker is undefeated in his 24 professional fights, with 18 of those wins coming by a knockout.
"Anthony Joshua is in for a huge shock," Parker said. "A couple of months ago I heard him say: 'Why should I be worried about this little kid from New Zealand?'
"Well, now he's about to find out. And the world is about to find out whether AJ can really take a punch."