Parker recently learned his sparring partner, Italian heavyweight Guido Vianello, has been denied dispensation to enter the country to help prepare him for the fight.
They have appealed the decision but if that too comes back negative, they will look to postpone the fight.
"We will not put Joseph in the ring against a guy ranked No 6 in the world by the WBO without a proper build-up," said manager David Higgins. "We went through the correct process to have him designated as an essential worker but that was denied on the grounds his skillset is otherwise available in New Zealand.
"I don't know if they have anybody in that [economic development] bureaucracy with boxing expertise but that is simply not the case and we have appealed that."
Stuart Duncan, one of the top matchmakers in the southern hemisphere, and Kevin Barry, Parker's trainer, both wrote to the ministry explaining that Vianello's height, weight and expertise were perfect to help prepare Parker for Fa's challenge.
Vianello lives with Barry in Las Vegas and has been a part of Parker's camps in the past. The 26-year-old Italian nicknamed The Gladiator has a professional record of 7-0, winning all seven fights by knockout.
Parker's management say they are not looking for special treatment and Vianello had committed to the 14-day quarantine protocols upon arrival.
Duncan and Barry explained there was nobody in New Zealand capable of preparing Parker to sufficient standard, but the Government was not buying that argument.
"We're not coming out swinging to put pressure on," said Higgins, "but we want common sense to prevail. We've seen dispensations made for the film industry, the America's Cup and other sports, so I don't know how this is different."
Indeed, the Parker camp could feel fully justified at the snub given the obvious health and safety benefits of having an appropriate sparring partner, as well as the economic ramifications of the fight not going ahead.
One significant sticking point that remains out of the promoter's control is Covid-19. If a crowd cannot attend, the fight will be postponed as the promoters would risk losing serious money even with a rich broadcasting deal.
If it goes ahead as scheduled you can make an argument that this could equal David Tua v Shane Cameron as the most prestigious all-Kiwi fight in history. While that was a huge hit locally, Parker-Fa would likely have more traction internationally as the industry has been starved of quality match-ups due to the pandemic.
Parker is 27-2 and ranked by the influential BoxRec.com ratings as the 12th-best heavyweight in the world. Fa is 19-0 and sits at No 31 on the same list.
While this is not part of Parker's three-fight deal with Matchroom, the Hearns will still use their considerable heft in the European and American markets to push this fight to a huge global audience.
A win by Parker, 28, would be a timely boost after his ascent plateaued following back-to-back losses to Anthony Joshua and Dillian Whyte. A win to Fa, 30, would move him to the fringes of title contender.
