Joshua successfully defended his IBF title, while also claiming the WBA 'super' belt at a sold out Wembley courtesy of a dramatic 11th-round stoppage, after the Brit had been flattened in the sixth.
Sky Sports reports Parker had been due to defend his own title against Hughie Fury in Auckland on May 6, but the unbeaten Brit was forced to pull out due to a lower back injury.
The 25-year-old Kiwi will now face Romanian Razvan Cojanu on the same date, but admits after that clash he will look to arrange something with Joshua.
"It's great to see Joshua. You know he went through a bit of a hard time in the fight and it's good to see him come back strong and maintain his title, and get the other ones," Parker said.
"I think Klitschko looked great. For 41-years-old he looked in great shape. He's still got a lot of power and I feel like it could have been his fight but I guess the better man won on the day.
"Joshua is the man to beat. He got off the canvas and showed heart but I think every fighter has things they've got to work on, weaknesses and strengths, and I feel like I just have to get past this next fight and then I look forward to sitting down and planning a fight with him in the future."
Parker has an unbeaten professional record of 22-0, and claimed the vacant WBO world title against Mexican Andy Ruiz in December 2016.
Joshua's professional record is now 19-0 and after being dragged into the longest fight of his career against Klitschko in front of 90,000, he could well be interested in trying to claim Parker's belt.
Photo: Anthony Parker celebrates his win against Wladimir Klitschko