Ramsey has scored 13 goals for the Gunners this season but has not featured since a Boxing Day win over West Ham due to a thigh injury.
The Wales international penned a new deal at the Emirates Stadium on Tuesday and Parlour reckons he can emulate the achievements of Liverpool and England skipper Gerrard.
"Without doubt, Aaron Ramsey can become as influential as Steven Gerrard. Ramsey needs to believe in his own abilities," he wrote on his Sportlobster blog.
"He is a young and exciting talent and will definitely be a top, top player in the future. He wants to learn and is blessed with the ideal attributes for a midfield player."
Parlour revealed Gerrard, whose Liverpool side are fighting for the Barclays Premier League title alongside the Gunners, leaders Chelsea and Manchester City, was the best opponent he came up against during his own playing days.
"Gerrard was the best player I have ever faced," he added.
"I think he is absolutely world class. He isn't as effective box-to-box as he was a few years ago, but he continues to make the difference.
"Ramsey was the perfect box-to-box midfielder in the first-half of the season and I always said if he could score a few more, he would be complete."
Arsene Wenger will take charge of his 1000th game in charge of the Gunners on Saturday when they travel to Chelsea.
With the two London rivals looking to make further progression towards the championship, Parlour believes a draw would be a good enough result for Arsenal.
"The match against Chelsea is absolutely massive for Arsenal and hopefully they will snatch a win, but a draw is a decent result also," he said.
"The spirit of the team is fantastic. You cannot win anything if the players aren't pulling in the same direction.
"Wenger is brilliant with man management. If you were out of favour, but always part of the team, he would always keep you involved and say you could change the game or even the season.
"He was very good at keeping you focused on the task. This is an important factor and his squad is very supportive of each other."
A win for Arsenal, who have a game in hand over Chelsea, would take them to within a point of the current leaders as Wenger continues the search for the club's first trophy in nine years.