The Frenchman was drawn as a potential candidate for the Three Lions job after Sven-Goran Eriksson claimed that he would be a perfect manager for the national side.
Despite the vote of confidence, Parlour believes that the 61-year-old belongs within club football.
"Arsene Wenger could definitely do it, but I don't think it's up his street," talkSPORT quotes Parlour as saying. "Every single day he's on that training field, he's a club manager.
"He likes that day-to-day involvement in the club, a little bit like Sir Alex Ferguson, he's in that office at 7am every morning.
"I don't think he'd look at it and say, 'this is the challenge I really want'. He's a club manager."
Meanwhile, the Gunners boss has conceded that the Premier League title is out of reach after seven games.