"More money for this fight than your whole career put together. Show some respect #Chump," Fury wrote.
Strictly speaking, it is Fury, who has remained mandatory challenger through the benevolence of the WBO despite pulling out of his first scheduled fight, with the opportunity of a lifetime.
The Englishman, who will earn well over $1million from the bout, has never been in the ring with a fighter of Parker's quality, something which Parker himself acknowledged.
Parker put the social media swipes down to a lack of confidence. The pair were friendly in Manchester last month on a promotional tour, but the tone is likely to be very different next week when they face each other again.
"Maybe that is due to a lack of confidence," Parker said. "I get my confidence from training and preparing well. Some people get theirs from talking. I know I have what it takes to win. I've done it before many times. He has never been in the ring with someone like me.
"I don't think he backs his power. He will try to hit and move and be awkward and I will be trying to knock him out - no question."
But while Parker has taken a philosophical attitude towards the outbursts, he agreed "there will be an extra edge" when the pair face off in the ring, with the Kiwi making what for him is a relatively rare statement of wanting to knock Fury out.
"I am disappointed with the promotional side of things," Parker added. "One of my goals when agreeing to fight in the UK was to raise my profile here and I'm not sure that will be the case - no matter my performance. If you look at the way Duco promote my fights at home - there's no comparison."
Parker's heavyweight rival and mate David Haye, meanwhile, has suggested that an all-out assault from the opening bell might not be the best option.
Haye, at whose London gym Parker is training, said: "Hughie Fury's got a very awkward style, a very under-rated defence, and it could sort of pose Joe a few tricks early-doors until he gets to grips with things so it's all about patience. [It's] all about him not getting frustrated in the heat of the battle and keeping a cool head, that's the best advice I can give."