Parker, 24 and undefeated after 20 professional fights, will meet Ukrainian Dimitrenko at Manukau's Vodafone Events Centre, the venue where he earned his shot against Joshua with a hard-fought victory over the tough Carlos Takam in May.
Price told World Boxing News he was impressed with Parker's win over Takam, saying: "I've followed him from early on when he was fighting the usual suspects just racking the wins up, but Parker's win against Takam stands out as Takam is a serious contender.
"Parker seems to have momentum, he's young and has fast hands but I do believe if I fight him I'll knock him out.
"I don't think he's been hit properly yet and obviously, my power is my main asset. He's a big puncher and it will probably be the first fight I go into as the underdog so the pressure is off for me, really.
"We've agreed the [Parker] fight, and as far as I'm aware, it's just a case of Pulev taking the Joshua fight or agreeing terms. If it gets to the point where the fight's on, then I'll get the Parker fight.
"But if Pulev doesn't agree to the Joshua fight then I believe Parker will step into the main fight. Obviously, I'm hoping they come to an agreement with Pulev. We're still in limbo a little bit at the moment. We just need to find out what's going on with the Pulev situation and we can go from there."