The governing body has ordered a bout between the undisputed heavyweight world champion and mandatory challenger Parker and the New Zealand fighter's promoter David Higgins has revealed that his charge wants to take on Usyk in a unification bout.
He said: "Joseph is hoping and praying that Usyk will want to defend the honour of undisputed champion, which not many people in recent times have achieved.
"Hardly anyone has even held that honour for more than a few months so we are hoping that Usyk will want to have a reign as heavyweight champion and will honour a unification against Joseph over a splinter of the belts."
The promoter added: "We haven't given up on Usyk fighting Joseph next. A lot of people are suggesting that he won't but for me that is a bit of a soft outcome if he doesn't.
"We are hopeful of fighting Usyk and we are hopeful of doing it before the end of the year."
Usyk became undisputed heavyweight world champion for a second time after knocking out Daniel Dubois and named Parker on a list of potential opponents for his next bout once he has rested for a couple of months.
He said after the fight: "Nothing is next. I want to rest. My family, my wife, my children, I want to rest now. Two or three months, I want to just rest.
"Maybe it’s Tyson Fury [next]. Maybe we have three choices, Derek Chisora and Anthony Joshua. Maybe Joseph Parker. Listen, I cannot now say because I want to go back home."