Parker admits he would give Carrick the job long-term, with his current caretaker deal expiring at the end of June. The former England fullback says Carrick cannot be compared to former manager Ole Gunnar Solskjaer's situation due to the difference in the players at his disposal.
He explained to Tribalfootball.com on behalf of Spreadex Sports: "I'd never say it won't happen again in football because football's going the way football is. You used to be able to say something like that because there'd be stronger people about. But the Solskjaer one was just the players he had.
"He just had so many bad apples, which had been left behind. Michael Carrick now is coming when a few of the contracts have ended. People are going. I think he needs to be able to be strong and allowed to be bold enough to make big decisions himself.
"If people want to go and offer him players, then allow him to choose the players that he thinks will fit the mold."
Indeed, Parker says it's important United target players who want to play for the club.
He added, "When you look at Mbeumo and you look at Cunha they made their point at other clubs and they wanted to come and play for Manchester United and it's as simple as that. That's the way it should be.
"Don't look at someone if they're weighing you up against someone else. You really don't want that person you let them go and say 'good luck to them' and then you move on."