"He hasn't fought in six months, but miraculously, at the age of 39, he's visibly got himself in terrific physical condition.
"As Joe's trainer, I need to be asking the question. As long as Minto is doing things the right way and doing it clean, then I have no problems."
Minto, who has been training under coach Chris Martin since arriving in New Zealand recently, said he went straight back to the gym with a renewed motivation after he beat Cameron in a fight where he was coached by Barry.
"I'm not too worried about it at all," Minto said of the tests. "They're just taking my hard work and trying to make it seem like it's not hard work.
"It's just Kevin Barry trying to play his mind games and everything so I'll just let them do whatever they want."
Duco's Dean Lonergan confirmed the testing would take place to ensure a level playing field.
"We know 100 per cent that our fighters are clean," Lonergan said. "So we just want to make sure everyone else is clean as well."
This is the second sticking point of the fight after Minto had a clause written in to his contract barring the use of the Everlast MX gloves, which are Parker's preferred mitts.
The MX gloves, made of horse hair and foam as opposed to a more traditional moulded foam, offer assistance to a heavy hitter like Parker (8-0) who has used the gloves in his last two knockout victories over Afa Tatupu and Nascimento.
Minto, ranked 12th by the WBO, will put his WBO Oriental heavyweight title on the line in next month's bout, while Parker's interim PABA crown will also be up for grabs.
Meanwhile, negotiations are ongoing to put Kiwi light heavyweight Robert Berridge on the big stage with a fight in the US on August 9.
Berridge (23-1-1), who will meet Brazilian Rogerio Damasco (12-1-2) on the undercard of Parker's bout with Minto, recently missed out on the opportunity to challenge WBO champion Sergey Kovalev later this year.
Lonergan said Berridge could fight on a card in Pennsylvania that will be headlined by Ukrainian heavyweight Vyacheslav Glazkov who is ranked No 2 by the IBF.