The Chicago performer, who is perhaps best known as the best friend of Robbie Williams, implied that he was bored of The Wanted on the daytime television show, saying: "God the wanted on loose women!!! Full of personality!!!! Me sarcasm never!!! (sic)".
Parker later replied: "What are you most famous for again. Hanging out the arse of a superstar? Looking forward to your next career move!!"
Wilkes quickly responded: "I'm in the west end at the moment playing Billy Flynn in Chicago, it's my 9th lead in a musical!
"Then I will go home to my beautiful wife and baby boy!!! Which we run 4 very successful academy's!!! So why don't you come and see me and make that comment to my face!!!"
However, Wilkes later backtracked on his tweets by suggesting that he is actually a fan of the boyband's music.
"Oh and for your info I like the wanted you make great pop music!!" he wrote. "I was just having banter!!! You see my kids at the academy love you!! And I was winding them up!!!!"
The Wanted recently attended a camp in order to improve their songwriting skills.
Parker previously revealed that he was "shattered" after being rejected from The X Factor as a teenager.
Listen to The Wanted talk about their new album at T4 On The Beach below: