"For those British contenders, respect to the one that does step up to Parker, because I think, despite Parker not quite getting the win, he gained a lot of respect, and is the first man to take Joshua the distance."
Parker is friendly with Haye, having used the British heavyweight's London gym while training for his last two fights in the UK against Joshua and Hughie Fury.
Another British fighter touted as a potential opponent for Parker is rising heavyweight Dillian Whyte, who has called out Parker in the past.
Higgins said Parker's performance against Joshua should silence his British critics, particularly Bellew and Whyte.
"A lot of guys are very mouthy and derogatory towards Parker in putting him down, like Bellew or like Dillian Whyte," Higgins said.
"It would be very interesting to see if they are still calling him out, and if the likes of Dillian Whyte or Bellew have changed their tune. What does that say about Bellew's character? What does it say about Dillian Whyte and his character?
"From our perspective, we are ready, willing and available. We're not going to chase it, because our view is that they were just all talk. Any fight against Parker now will be a guaranteed sellout of the O2 against any of those guys. Credit to the one that steps up."
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