The Gunners, who are currently fourth in the Premier League, are locked in a battle to secure Champions League football, while also having the opportunity to end a nine-year trophy drought in the FA Cup.
Ahead of the FA Cup semi-final against Wigan Athletic on Saturday, former Arsenal midfielder Parlour believes that Wenger will refuse to sign a new deal if he thinks he can't improve the team.
"It's such an important week for Arsene Wenger as a manager," the 41-year-old told the Daily Star. "He looks like a broken man sometimes.
"How many big games now have some of his players failed to turn up in? He will be trying to put that right.
"But if he thinks he can't take the club forward in the summer - he will walk away. He's in a situation now in his career where he's a bit older and he has his health to consider."
Wenger hinted on Thursday that he is starting to line up potential transfer targets.