Adding intrigue to the Fury sideshow is the fact his rematch against Klitschko, delayed until October 29 due to an ankle injury suffered in training, has not been confirmed to the Ukrainian.
Fury, who has said the Klitschko rematch will be his last fight, partied with English football supporters during the recent European Championships in France despite his injury and upcoming bout.
"Might there be another reason for calling it off?" Klitschko said recently. "Is the delay supposed to unsettle me? Or in [Fury's] case, is it manoeuvring on the part of your negotiating partner that he stands you up twice and then wants to meet you at a late hour?"
Fury, aware Klitschko likes to be in control, unsettled him before their fight in Dusseldorf by refusing to accept the ring surface. Fury wanted a harder surface and it was changed at the last moment.
Parker is the mandatory challenger to Joshua's title, but the Englishman is likely to fight Bermane Stiverne in London in November, which means a Joshua-Parker fight probably won't happen until March or April. Joshua's promoters, aware their man is a huge money spinner but still relatively inexperienced, are likely to delay the Parker fight as long as possible.
The major upside to a Klitschko fight for Parker would be the money and the fact he is ranked higher than the Ukrainian by the WBO. The 24-year-old could claim a 50-50 purse split if he fought 40-year-old Klitschko, whom he has sparred against. Parker would likely get 25 per cent of a Joshua purse.
Danny Leigh, WBO's Asia-Pacific executive, told the Herald on Sunday his organisation would make a decision on Fury once all the facts were known and confirmed a Parker v Klitschko fight to determine the championship is possible.
Should Klitschko retire, then undefeated Mexican-American Andy Ruiz, ranked No 3, would fight Parker for the championship, Leigh said.
Parker sparred against Klitschko in Florida last year and apparently performed well against a man who turned professional in 1996 and has lost only four of 68 pro fights.
"He's extremely talented," Klitschko said of Parker following the camp. "He's strong but he is very different from Anthony Joshua, even though they are both very determined and very tough. I see myself in Joseph, in a certain way. We've had some intense sparring. He has given me great work, exactly what I need."