The Portuguese playmaker quickly established himself as a star at Old Trafford after joining from Sporting Lisbon in January 2020.
Pogba's form has been exceptional in recent weeks, and Parker thinks it could have something to do with Fernandes.
"In 2020 he was been playing in the shadow of Fernandes and that will have dented his ego," The former Red Devils defender told Eurosport.
"Pogba was signed for a big fee; his marketability and profile - as well as his playing ability - suited Manchester United.
"But the best profile you can ever have is to be an excellent footballer, and while Fernandes came in with a much lower profile, his status has outgrown Pogba's and for the right reasons: he's at the top of the tree for being brilliant.
"And you can't beat competition. In any industry, if competition comes along and you just sulk about it, then you're not going to survive. You have to improve to get respect.
"Fernandes quickly became the key man when he arrived and it became Manchester United and Fernandes, not Pogba. It was Pogba before for the wrong reasons, now it's Fernandes for goals and assists.
"United need that from Pogba now to challenge for the title, for him to not drop below a 7/10 in any game."