Fa defeated Argentine journeyman Rogelio Omar Rossi in 86 seconds on the undercard of Parker's win over Alexander Flores at Horncastle Arena in Christchurch on Saturday.
The win takes the 29-year-old's record to 16-0, as he continues to work his way back to full fitness, after overcoming career-threatening health issues.
Barry revealed that Fa had initially been offered $500,000 to fight Parker, but his management declined the all-New Zealand affair, so Duco then approached Flores, who gladly accepted.
Parker crushed Flores to win by knockout in the third round, but the 26-year-old Kiwi was warned twice for throwing low blows in a controversial fight.
Despite that, Parker's camp was happy to be back in the winner's circle, after losses to Anthony Joshua and Dillian Whyte this year.
Speaking after the win, Barry was quizzed about his fighter's options for 2019 and Fa's name was mentioned.
But Barry didn't want to talk about the fellow Kiwi, as he unleashed a scathing attack at the WBO Oriental champion.
"We're here celebrating a great victory, fighting the best fighters in the world, don't ask me a question about Junior Fa," he said.
"When he fights somebody inside the top 200 - I don't say that as a joke, look at the records, look at the names, look at their BoxRecs - they need to start testing him.
"If we're ever going to get a fight with him, he needs to fight somebody who's good. It's a joke he gets mentioned in the same breath with Joe.
"We offered him this fight, but he said, 'I'd rather fight a guy who has had seven fights in seven years and lost four of his last seven fights by knockout, and he's 37.' Give me a fricken break."
Barry then said he wouldn't answer any more questions about Fa, adding they had many other options on the table.
One of them is a trans-Tasman bout against Australian Lucas Browne (27-1) in April next year, but Parker still has his eyes set on a rematch against Whyte, according to Barry.
"We'd love to fight Dillian Whyte again," added Barry. "That is something Joe talks about on a weekly basis.
"Dillian has a big fight next week against Dereck Chisora. We'll fight the winner of that fight or the loser, but we have options.
"We've also sat down with Eddie Hearn and we have some options in the US."
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