Parker, the World Boxing Organisation champion, makes his maiden defence against Fury at Auckland's Vector Arena.
NZ Herald reports Joshua, the International Boxing Federation titleholder, and Klitschko, the former IBF, WBO and World Boxing Association champion, fight for the vacant WBA belt at London's Wembley Stadium.
A total of 90,000 tickets are understood to have been sold for the latter fight alongside an expected million pay-per-view passes, courtesy of an exclusive deal with BSkyB. Budgets suggest the fight will generate more than 50 million in revenue.
If Joshua beats Klitschko, his next bout is likely to be in November or December.
"We [would] have two belts," Hearn said. "We want all the belts and Joseph Parker has one. You have Deontay Wilder [the World Boxing Council champion] and you have Parker. I think the match-up, style-wise, between Parker and Joshua is exciting.
"If he can come through Hughie Fury, we can look at a potential unification fight [for three of the four major belts] later in the year."
Hearn said a Parker victory was crucial for his profile in Britain.
"This is a solid first defence, it's a mandatory [defence] for him as well. He has to go out there and look good.
"People want to see knockouts. Parker is a similar type of fighter to Anthony Joshua. He's young, athletic and explosive, and that's what people want to see. They don't want to see 12 rounds of exhibitions [with] people grabbing and holding.
"They want to see fighters who let their hands go. Joshua and Parker do that.
"The Brit in me wants to see Hughie Fury do the business. But the promoter in me looks at a potential match-up between Joshua and Parker, and I can't help but lick my lips."
Parker might also benefit from what is perceived as Wilder's lack of profile in the United States.
Four of the American's five WBC defences occurred in his home state of Alabama.
As a result, luring money from the big US television networks is understood to have been difficult.