Solskjaer has won eight of his first nine matches in interim charge of Manchester United.
The one blemish came on Tuesday evening when the Red Devils came from two goals down to drawn 2-2 at home to Burnley.
In early January, four games into Solskjaer’s reign, Parker claimed the Norwegian should not be given the job full time.
He said : “Long-term, I don’t think so. As good as it can be this season I think you have to look at the long-term.
“Look at Manchester United, they need that little bit more. Someone who’s really going to go out and attract the right players.”
But now Parker says he can see Ole Gunnar Solskjaer managing his former club for years to come.
However, he says United must appoint a sporting director to help him in his role.
“Initially on this show I turned round and said about Ole that I worry about him attracting players,” Parker said on Wednesday.
“He’s a legend with Manchester United but I was worried about him attracting players to Manchester United.
“But what he’s gone and done says to me he has definitely jumped up high [in the running for the job].
“I believe he could manage Manchester United next season and beyond.
“But I think he’s got to have a director of football, a sporting director with him.
“He’s already got the godfather [Sir Alex Ferguson] behind him already. The boss is involved, he’s there, definitely there.
“And maybe a sporting director, somebody who works with all these agents who knows this man and knows the line he’s going down.
“I think he can do that job if he’s got the right people around him.
“We’ve seen it already, he’s got Mike Phelan next to him, people see him sitting there looking quite relaxed but he’s got a major say on what’s happening on that field of play.”