The former France international was a teammate of Parlour under Arsene Wenger between 1996 to 1999.
Villa are rooted to the bottom of the Premier League table with just eight points from 20 games and look certainties for relegation.
This weekend Garde's side face League Two promotion hopefuls Wycombe in the third round of the world's oldest cup competition.
Many fans and pundits feel Villa could do without the distraction of a cup run.
'Nothing to lose'
But speaking on talkSPORT, Parlour has offered Garde the opposite advice.
"I think Villa should go for it, what have they got to lose," Parlour told talkSPORT.
"They could get a win and that would be great for confidence wouldn't it?
"I think Remi should put his strongest possible team out and try and get a win, get a win for the players and the fans, get that winning feeling back.
"They could get a few goals, the strikers could get some confidence too.
"I think Villa should go for it, why not, they have got nothing to lose."
Last win was in the cup
The last time Villa tasted victory was in the cups, way back in September thanks to a 1-0 win over fierce rivals Birmingham City in the third round of the League Cup.
What followed was a six game losing streak, though, which does not give much weight to Parlour's argument.
That was before Garde took over, though, and Villa have looked a better side since his appointment, even if results haven't improved.
But even if it is just to give the suffering fans of Villa some new year cheer, a cup run could be a welcome distraction from the gloom of the league.