Some of the names putting their hands up would be a good fit for Parker as he seeks to go to a new level in terms of his performance.
A bout against Japanese heavyweight Kyotaro Fujimoto at, say, the Tokyo Dome would be a good one for Parker. Fujimoto is a 31-year-old with a 17-1 professional record ranked ninth by the WBO and 14th by the WBC. His loss came against Solomon Haumono, the Australian whom Parker walked through in four rounds, which suggests a relatively easy night's work would be in store for the Kiwi.
Would it be a significant victory for Parker? Probably not for his wider development, but it would be a good payday and a step in the right direction in terms of stoppages.
Australian Lucas Browne would also appear a good fit, as would Dillian Whyte from England, although a low-ball offer from his promoter Eddie Hearn and the fact that Whyte could be set to fight American Dominic Breazeale next month would appear to rule that one out in the near future.
Parker is keen to fight again between November and the New Year.
Both men would stand in front of Parker and engage - something Fury refused to do, which contributed to an ordinary bout at the Manchester Arena last month.
A potential fight against Russian Alexander Povetkin would set off alarm bells.
Povetkin is a genuine heavyweight force, a man who has lost only once in 33 professional fights (by decision to Wladimir Klitschko), a man, too, who knocked out Carlos Takam, a division gatekeeper whom Parker could beat only by decision.
A fight in Moscow would present its own challenges and risks.
It would be foolish to read too much into the recent comments by Hearn, Joshua's promoter and also the man behind Whyte and Tony Bellew, when he told Seconds Out Boxing: "Now people think Joshua against Parker is a mismatch. I don't necessarily agree with that.
"Stylistically, it's a good fight, and he's got a belt. But he's very difficult to promote [in the UK]."
Hearn clearly has his own agenda.
Parker is a drawcard for as long as he retains his WBO title, but with Jarrell "Big Baby" Miller beginning to look formidable in the United States and Daniel Dubois looking powerful and explosive in the United Kingdom as he begins his professional career, it's time for the Kiwi to look dominant in the ring again.