Klitschko, who retired after being knocked out by Joshua in their April fight at Wembley, says he is comfortable with his decision to step away from the boxing ring and believes Joshua is the fighter to carry the glamour division forward.
"I have passed the torch to the younger lion," Klitschko told The Sun.
"I wish, from the bottom of my heart, he continues to carry this torch for a long time and is very successful just like I was. Joshua is capable of unifying the division, absolutely."
Joshua, who added the WBA belt to his IBF title by beating Klitschko, is in serious talks with WBO champion Parker for a unification fight. It's looking likely that will take place in Britain in March.
Joshua has also targeted WBC champion Deontay Wilder as he looks to hold all the major belts.
"I'd say the more Joshua shines, the better I shine. If I lost, I want to make sure I lost to the better man," Klitschko, 41, said.
He is happy in retirement though he still believes he could be a force.
"I've still got it. It's all about perfect timing It was my most complicated decision and took me 100 days. No regrets," Klitschko said.
"I never thought I would be the dominant force for 11 years non-stop and the second longest-serving heavyweight champion in history. I know I didn't waste my time."
Klitschko refused to discuss the return of Tyson Fury who beat the Ukrainian in December 2015 to rip open the division. Fury then spiralled into personal problems that have seen him sidelined ever since, including failing to make to scheduled rematches with Klitschko.
"Listen. I don't want to bring any attention to him - it doesn't make sense," Klitschko said.
Meanwhile, Klitschko raised the equivalent of NZ$300,000 for charity by selling the black robe he wore into the ring for his fight against Joshua.
The robe had a USB stick sewn into it that contained his prediction for the fight.
The robe was auctioned at a charity event at London's Royal Horticultural Society Halls, and the money will go to projects supported by the Klitschko Foundation.
Photo by: GETTY IMAGES. Caption: Anthony Joshua and Wladimir Klitschko after their epic heavyweight fight in London in April.