Parker, who earned his No1 challenger spot with a hard-fought 12-round victory over Carlos Takam in Manukau, has returned to his Las Vegas base after a break and is about to resume sparring.
Trainer Kevin Barry said Parker's elbow soreness would be carefully managed during this time because it was during sparring when the problems generally arose. After the Takam fight, Parker revealed his body "broke down" during the tough 12-week camp, and said the pain in his left arm robbed it of some of its venom and added to his defensive issues.
Barry said today: "We've had a niggly elbow for a couple of fights now that we've been managing. You probably saw me icing it during the Takam fight. Whether or not that was the reason his hand was so low, I'd like to give him the benefit of the doubt and say yes. But I definitely want the hands up a lot higher in the next fight.
"At the moment it's not an issue in training. It usually doesn't flare up until sparring. We are doing a lot of neuro-muscular work on it.
"The thing with Joe is that he has such fast hands and if he punches at that sort of velocity and he doesn't hit the target, there's going to be some hyper-extension."
Barry said he and Parker would not travel to London to watch Joshua's first defence of his title, but would watch it closely.
"I've been very impressed with [Joshua] during his last two fights. I think he has developed and improved a lot during his last two fights.
"He's a big guy and he uses the height he has very effectively. I'd be lying if I said we wouldn't be watching him intently, because we will be, just as I'm sure he would have been watching Joe closely in his last fight."