Tua is the godfather of Barry's twin sons. However, Barry admitted the pair now had "no relationship whatsoever".
"It really didn't end well and it was really sad. We had such an incredible journey together - 12 years of incredible success and incredible experiences.
"Sure, it ended really, really ugly. But David was a huge part of my life and part of my family. He was a godfather to my boys. We accomplished so much together."
The 53-year-old Barry now has a chance to play a formative role in Parker's journey. A 21-year-old with a 4-0 professional record, Parker will stay at Barry's house and train at Barry's Boxing Gymnasium - which is owned and run by friends of the same name.
The arrangement represents a return to fight training for Barry after he closed his own Las Vegas gym following ankle replacement surgery.
His clouded history with Tua was no reason for concern when it came to Parker, he said.
"If anyone had reason for concern you think it would be Joseph's parents, but they were so warm towards me and appreciative and thanking me for the fact I am working with their son. They never mentioned David Tua's name."
Duco's David Higgins said Tua had appeared relaxed when told of Barry's appointment.
"I would say it was a non-issue," Higgins said.
Barry's record with Tua was in fact a big part of the reason for the appointment, Higgins said.
Linking with Barry would provide Parker with high quality sparring that he couldn't access in New Zealand.
Parker is close to appointing new management after splitting with Sir Robert Jones when Jones and Duco were unable to agree on suitable opponents.
The Herald understands Parker is in negotiations with prominent rugby player manager Bruce Sharrock.