The 28-year-old Briton holds the IBF and WBA belts, while the 26-year-old New Zealander is the WBO champion.
Whoever wins in March is certain to then face undefeated American Deontay Wilder, who holds the fourth belt in the heavyweight division.
"I would like to announce the official news that myself and Joseph Parker will be fighting on March 31 at Principality Stadium in Cardiff," Joshua is quoted by The Telegraph as saying.
"It is a unification heavyweight championship fight, we all know what happened last time I was in a unification heavyweight championship fight. It was gruelling, it was interesting and we both left the ring with masses of respect.
"These fights aren't easy because there is a lot on the line, so respect to team Parker for taking the challenge. And you know me, I love this game. I am looking forward to it, training camp is underway and before you know it March 31 will be upon us. Stay tuned for more news and I will see you all soon, God bless."
Both Joshua and Parker boast unbeaten records in professional boxing.