The episode is set to screen in the United Kingdom this weekend and it shows Parker defending his camp's tactics of highlighting what they say is Joshua's vulnerable chin in order to secure the fight.
It is hosted by former British cruiserweight Johnny Nelson, now a broadcaster, who asks Joshua what it feels to be sitting in front of Parker: "This is different," Joshua says. "Fighting ... it's going to be straight blood, last ounce of energy because there's two fighters that don't want to give in, so that's the fighting terms.
"In terms of securing the fight, he says what he says and does what he has to do. In terms of this, it's new and I think, as I said, I respect the family side because I'm a family man, I respect that all day.
"But this is the time to let the world know who you are, how great you are and it doesn't matter what he said about me, I just think you have to tell the world that you are the best because they are listening and this is history now."
Asked again about the way Parker's promoter David Higgins talked about Joshua's "glass jaw", and whether the Kiwi could be "punished" for it in the ring, Joshua says:
"I'll punish him anyway, it's not about what his promoter said, I'm going to have to punish him anyway, it's the punishing game."
Read more: Boxing: Team Parker – Wilder win 'good result' for heavyweight division
Nelson asks Parker to describe how the fight will play out.
Parker, his voice steady and his eyes on Joshua, says: "The fight will be a very exciting fight."
"Why," asks Joshua.
Parker: "Because I'm prepared, I'm going to come ready, I'm going to be in the best shape and I'm sure you'll do the same. But you say you are going to punish me, it's not going to happen because I'm going to chase you around ... you come at me with your power, I'll give it back."
Joshua interjects: "I'm going to meet you head on."
Parker says: "I'm looking forward to it."
The leaked exchange is likely to result in a big audience for the half-hour interview and will spark more debate among boxing fans as to who will win the world unification fight.
Read more: Kick-boxing cancer survivor calling the shots for Joseph Parker
Joshua is looking slim and quick in clips posted on social media from his base in Sheffield, while Parker looks in good form at his camp in Las Vegas.
Parker will arrive in London to continue his preparations a fortnight before the fight at the Principality Stadium.
To get the day's top sports stories in your inbox, sign up to our newsletter here