Parker, whose title unification fight against Joshua has been confirmed for April 1 (NZ time) in Cardiff, told the Radio Sport breakast show that being called the king of pies by the Joshua camp didn't worry him.
''They call me the king of pies but I call him the king of steroids. Who knows, if you are that big and that musclelly there's something wrong.
''I have heard word; I have looked at him and I have seen. He has been very close with Klitschko ... he has been a reigning champion for a long time. So I don't know ... it may be true maybe not.
Asked how he thought Joshua would respond to the drugs suggestions Parker said: ''I don't know how he will he respond ... he may be angry or upset at what we have been saying. It's part of the game. I am not accusing him of using steroids but I am saying it is a possibility.''
Parker said there would be drug testing prior to the title fight in Wales but there ''there are a lot of ways you can dodge drug testing''.
Months of negotiations ended today with the announcement of the fight, the first clash of two reigning heavyweight champions on UK soil.
The fight was confirmed by Joshua's promoter Eddie Hearn, who tweeted it would take place at Principality Stadium, a venue more famous for rugby internationals.
Parker, the first New Zealand-born heavyweight champ, immediately continued the confident talk that has dominated his team's public approach to fight negotiations.
"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.
"My entire existence is now devoted to proving what the boxing world already knows."
The pair are regarded as the premier young fighters in a division which is undergoing change following the retirement of long-time champion Wladimir Klitschko.
Parker, 26 and boasting a 24-0 record, is the WBO champion after beating Mexican Andy Ruiz in late 2016. He defended the title twice in underwhelming style last year, failing to knock out Romanian Razvan Cojanu and Briton Hughie Fury.
London Olympic champion Joshua, 28, (20-0) will defend his WBA, IBF and IBO belts, secured when he knocked out Klitschko in a compelling win last April at Wembley Stadium.
He defended the titles in a tough clash in Cardiff in October against French veteran Carlos Takam, a fighter Parker has also worked hard to overcome.
Joshua referred to his Klitschko win when looking ahead to the challenge posed by the smaller Parker.
"We all know what happened last time I was in a unification fight. It was gruelling, it was interesting, and we both left the ring with masses of respect," Joshua said.
"These fights aren't easy, cause there's a lot on the line, so respect to Team Parker for taking the challenge.
"You know me - I love this game - I'm looking forward to it, training camp is under way, and before you know it, March 31st will be upon us."
The winner will go within touching distance of becoming the first undisputed heavyweight champion of the world since Briton Lennox Lewis in 1999.
American Deontay Wilder defends his WBC crown against highly-rated Cuban Luis Ortiz on March 3.
