But he added the murky world of professional boxing wasn't for the faint-hearted.
"We wind each other up," he said. "It's the norm in boxing, there is a lot of banter. A lot of people get upset with the things I say but it's not what I mean at all."
He now understood where Higgins was coming from, but didn't appreciate the way the New Zealand promoter chose to get his views across.
"Well, I wasn't too happy with it," he said. "But it was done on the spur of the moment and it was a bit of anxiety from David trying to get the best for his fighter.
"I suppose he thought the only way he could get success was by doing what he did."
Talking to the Weekend Herald from the UK, Higgins said his press conference antics were "deliberate", both to get exposure for the world title fight and also have Terry O'Connor replaced as ref; something that was confirmed on Thursday.
"My position was completely legitimate," he said.
Higgins said he was happy to take reaction to his actions on the chin despite being this week's "walking headline".
"In boxing you have to have a thick skin" he conceded.
"When I was younger I used to care but once you've been through so many battles you figure what's important in your life - you know, family, friends, loyalty. You can't control what other people think."
And he still stood by his comments at the press conference.
"I make no apologies for fighting [for] a level playing field for Joseph Parker," he said. "I make no apologies for highlighting the hypocrisies of the British boxing board in trying to get an impartial referee. I make no apologies for fighting for some equality on the map.
"People know about the fight, they will watch it and talk about it and follow Joe's career. Have I ruffled a few feathers? Probably. But I make no apologies because my heart is in the right place and I am doing my job."
Fury said he made his comments to Higgins to get a rise out of him.
In relation to his son calling Higgins "Snow White", he said that was in relation to Higgins appearing "very pale".
"David looked very stressed and pale - like he'd seen a ghost," he laughed.
Peter Fury is one of the most colourful figures in the UK boxing scene.
He has previously served prison time for possession and intent to supply drugs, as well as money laundering.
His criminal past almost saw him barred from entering New Zealand for an earlier planned, but subsequently canned, bout between his son and Parker.
"I don't hide my past but everybody has a past - so everybody should own up to it and accept what they do," he said. "I am not proud of my past - it is regrettable.
"I'd like to say 'Hello' New Zealand. You have a lovely country. I haven't been. I'd love to come but I don't think I'm allowed."
Meanwhile, Higgins confirmed his partner Augustine Mon is in the UK with him.
Mon split from her husband ex-All Black Marc Ellis last year.
"We are good," Higgins said of his relationship. "All I can say is she is wonderful and bloody supportive."
He is also confident Parker will successfully defend his world title on Sunday.
"If Joseph does box Fury out - that's what people will remember and one day they might look back and say, 'David helped make the fight famous and put him on the map'."
