O'Connor was the sole judge of the contest and handed Fury the decision 98-92 despite the boxer's corner telling him he was behind on points before he went out for the 10th and final round.
Respected boxing writer Kevin Mitchell described the result as a "highway robbery" and the BBBoC investigated but took no action.
Last year, O'Connor was criticised for not stopping earlier a fight between George Groves and Eduard Gutknecht which left the latter in a coma.
In sending Duggan to England as fight supervisor, the WBO has signalled its intent and the organisation is likely to come under increasing pressure to do more over the next few days.
"Part of me says it is heart-warming to have the backing of the WBO and to see them taking an interest in their heavyweight champion," Higgins said. "This is about Joe being treated fairly."
Parker said: "It is important when you fight that you are on an even playing field but it is not my job to worry about the referee. My job is to be in great shape and prepare myself for fight night. I'm going to focus on what I have to do."
Bob Arum, the veteran American promoter who now has a business relationship with Duco and backs Parker's interests in the United States, has close links with the WBO and is understood to be fuming about the selection of the officials.
Under British boxing rules, the BBBoC has the final say over the selection of the officials. General secretary Robert Smith has said: "As far as we're concerned, we've had a bit of correspondence from Parker's team. We have heard nothing from the actual promoter in this country, Mick Hennessey, but we certainly had something from Parker's team, giving us their thoughts.
"We take on board what their thoughts are, we don't agree. We feel on this occasion, there is no need to deviate from our standard policy."
Kiwi Parker is in London preparing for the fight and will travel to Manchester on Thursday. He has already said his goal in the second defence of his WBO world title is to knock out Fury, a finish which would take the judges out of the equation.