Wenger responded to Chelsea manager Mourinho's taunt - relating to the Gunners' trophy drought dating back to the 2005 FA Cup - by saying he was "embarrassed" for the Portuguese.
And former Arsenal midfield mainstay Parlour, who was speaking at a Vauxhall Football Mash Up event in Paddington, told Press Association Sport: "Arsene Wenger won't let it bother him. He's had his say back and that's the end of it.
"That's how Arsene Wenger works. He'll be worrying about his team and getting them prepared for Wednesday night against Bayern Munich.
"It's a much bigger deal than what Jose Mourinho said."
The comments reopened a feud between the two bosses following a fragile truce and prompted Parlour to suggest there should be more respect shown between managers.
"I think there should be a code of respect, like when you respect your fellow players," said Parlour.
"Everybody's trying to win trophies, everybody's trying to be successful, but sometimes you can go over the top a little bit.
"Whether Jose Mourinho looks back - and he probably doesn't regret it - but maybe looks back and says 'Maybe I went too far this time'. If you look at Arsene Wenger's record it's not too bad: two doubles, he won the FA Cup a few times, he had an unbeaten season.
"Everybody knows the last eight or nine years they haven't won a trophy, they're trying to put that right.
"A lot has changed in the club. They've more or less paid the stadium off and there's lots of opportunity now."
Parlour, who played for six years under Wenger, hopes Arsenal can end their trophy drought this term.
As well as their Premier League and Champions League challenges, Arsenal have secured a home tussle with Everton in the quarter-finals of the FA Cup - a competition Mourinho's Chelsea exited on Saturday.
Parlour said: "I'm always very optimistic. I look at the FA Cup. It's going to be a tricky game against Everton, but they've got to capitalise on home advantage to try to get to Wembley for the semi-final."
First the Gunners must focus on playing Bayern, a team they beat in the second leg at the last-16 stage last season but still tumbled out to on away goals.
The German giants visit the Emirates Stadium first, and Parlour fancies his old club to gain a first-leg advantage.
"Arsenal will have a chance being at home," Parlour said.
"I just hope they can do themselves justice and put their grade A game on and give Bayern Munich a game."
Parlour was speaking at an event designed to encourage teenagers to keep playing football.
The Football Association's director of the national game and women's football Kelly Simmons told Press Association Sport: "Mini-soccer and children's football has been going through big growth, but adult football's been in decline.
"One of the things we know we've got to do to help grow the adult game is to stop that drop-off.
"Some of them just want to be able to turn up, play games and have fun when they can, more recreational.
"If teenagers want to play in our traditional club football, then fantastic.
"If they want something more recreational, more flexible, we've got that too."