The Gabonese striker has become the subject of unwanted transfer talk of late.
A shock switch to Old Trafford has been mooted, with the 30-year-old said to be open to the idea of heading north from the Emirates Stadium.
Dutch frontman Van Persie did likewise after deciding that he needed to leave Arsenal in order to land major honours and the fear is that Aubameyang could follow that lead.
Parlour believes a sale will be avoided, with the former Gunners star confident that lessons have been learned from the past.
He told talkSPORT: “It would be madness to let him go, but players have got all the power these days.
“Aubameyang and [Alexandre] Lacazette have been absolutely brilliant this season. It looks like they’ve got a really good relationship up front, you can see how they celebrate goals.
“I’d be amazed by that [if he signed for United], I really would. That would shock me.
“That wouldn’t be good from an Arsenal supporters’ point of view – we all remember when Robin van Persie left to United and they ended up winning the league the following season….
“When I go to pre-season, I’m really hoping Aubameyang is on that plane.
“Goalscorers are so important. Good ones are hard to find and he’s really proven he is a top class goalscorer.
“You want to be adding to your squad, not trying to get rid of a few players!”
Aubameyang proved his value to Arsenal in the 2018-19 campaign – his first full season in north London.
He recorded 22 goals in the Premier League for the Gunners, with that return enough to earn him a share of the Golden Boot alongside Liverpool duo Mohamed Salah and Sadio Mane.