Hearn has earned the ire of both Higgins and Peter Fury because of the Englishman's alleged undermining of the Parker v Fury promotion. The details aren't clear but it is alleged Hearn, the promoter of WBA and IBF heavyweight world champion Anthony Joshua and cruiserweight champion Tony Bellew, had been talking down the chances of the fight happening.
Hearn's sudden change of heart about meeting a touring party of Parker's supporters had also gone down badly with Higgins.
"This promotion and the Fury camp have proved a lot of people wrong," Higgins said. "A lot of big egos in English boxing rubbished them and said they wouldn't put the fight on and wouldn't put the money in the bank. They rubbished me for backing the word of Peter Fury.
"Guys like Eddie Hearn, who promised to meet our New Zealand travel group, but pulled out on the day, break their word.
"It's been a long and winding and tricky road putting this fight on but it's absolutely on.
"Tony Bellew as well ... he sat a foot away from me, I couldn't get a word in, he shouted at me for half an hour about how it's never going to happen because these guys wouldn't put up the money, well blow-hard Bellew has a bit of egg on his face and hopefully we can fight him also one day."
Higgins sparked laughter and applause when saying in the direction of Peter Fury, who spent nine years in prison after being found guilty of dealing amphetamines: "I'd just like to thank Peter Fury for saying I'm safe here in Manchester and I'll go home in one piece, because... if it was Eddie Hearn, I'd have my doubts. But Peter is a man of his word so we're all going home safe."
Higgins also fired a shot at critics in New Zealand, including Sir Bob Jones, a former backer of Parker's, when he said: "There are a few silly old know-it-alls back in New Zealand - and they know who they are - who are saying I'm out of my depth, that sort of thing. We wouldn't be sitting here today if I was out of my depth.
"The reality is I've been promoting boxing only eight years - I've only got two boxers [Parker and Australian welterweight Jeff Horn], and they're both world champions, so that's a 100 per cent record."
Peter Fury said: "A lot of these promoters and certain fighters thought he [Higgins] was wet behind the ears and tried manipulating him, telling him the fight's not going to happen. I would call you devious little bastards, because that's what you are... this is what's gone on behind the scenes, this is boxing politics."
In a joint interview with Higgins later, Peter Fury said of the Kiwi's antics earlier in the week: "He can't offend me for some reason because I do like him. He just makes me laugh. I know he's harmless in the sense that I know his heart is in the right place - he's just looking out for his fighter."
Higgins said: "To make an entrance, maybe I embarrassed myself a little, but I took one for the team and we got a new referee."
Earlier, Tyson Fury, the lineal champion after beating Wladimir Klitschko in Germany in 2015 but who was stripped of his titles - including the WBO belt which now belongs to Parker - due to his inactivity and admittance to using cocaine, turned up with a big entourage before taking a seat among the media and onlookers.
Spotting Parker, who arrived wearing a smart blue suit, Tyson said: "Joe, Joe! Looking sharp, Joe." The pair shook hands.
Asked by the Herald for an interview, Tyson politely declined, saying: "I don't do interviews these days, sorry. I say things that get me in trouble."
Parker v Fury at Manchester Arena will be shown in New Zealand live on Sky Arena from 8am on Sunday for $39.95.
