For whatever reason, the Englishman wouldn't look at Parker during their press conference to promote Sunday's world heavyweight unification title bout in Cardiff, but he stared down at the Kiwi throughout the face-off which came immediately afterwards.
Parker has a plan to deal with his size and reach disadvantage – he gives up 15cm in wing-span – and he was right when he said he likes fighting taller opponents.
"In the past the bigger guys have suited my style better than shorter guys," he said.
The challenge, as he admits, is big, and he will need a lot of things to go his way should he emerge with the victory.
"I see him as a great champion," Parker said of his opponent. "The reason I wanted this fight is because I respect him. We respect what he's achieved in the boxing world, we respect his team and what they've achieved on this side of the world. We love challenges.
"We see him as a big challenge which is the reason we want to fight him."
His confidence comes from a sound body and the knowledge that he couldn't have worked any harder or done any more in training. Parker's trainer Kevin Barry said: "This last camp, the last 11 weeks in Las Vegas, has been the most enjoyable time that Joe and I have had together in five years.
"I've said all along that of all the heavyweights in the world I believe Joe has the best skills. I believe he can do more. I believe he has the best hands."
Joshua is clearly miffed at Team Parker's aggressive approach to promoting this fight in Cardiff, and in particular David Higgins' refusal to concede any advantage.
In the UK's Times this week, Higgins was quoted as saying: "A lot of Joshua's opponents have bowed down and formed a master-slave relationship.
"They're offered a title shot by Eddie Hearn and I think they see it as, one, maybe the only title shot they will ever get and, two, the relatively small purse could be their biggest ever payday. They'll take it, be very polite, and not criticise Joshua.
"We're not here for the pay cheque. We're not in a position where we quietly have to thank them for the opportunity and take their first offer. Joseph is here to unify the division."
That might go some way towards explaining Joshua's muted response to the question about Parker's strengths as a boxer.
"As everyone says, he's quick and he can take a punch - that's what I've heard anyway," he said.
