The Magpies are continuing to build after becoming the world's richest football club, with the £305million takeover by Saudi Arabia's Public Investment Fund backed by Mohammed bin Salman who is worth an estimated £320billion.
Despite the substantial backing, Newcastle are so far avoiding overspending in the transfer market, and are yet to break their record signing fee while bringing in a number of new arrivals.
The takeover came in October last year with the club in the midst of a relegation battle and, after naming Eddie Howe as their new manager and a productive January window, safety was secured with an 11th-placed finish.
Much like Chelsea and Man City before them, Parlour sees a steady climb up the table starting next season, with a cup run key to success.
The Arsenal legend told talkSPORT Breakfast: "It could be a Man City, three or four years it took Man City to start bringing the right players in, to get the right manager, I think Eddie Howe will get an opportunity now.
"I think a cup run is going to be important for Newcastle, I think they need it, imagine them getting to Wembley and winning the FA Cup.
"That would be magnificent for the club so I reckon top eight for Newcastle next season and a good cup run.
"You want to see teams like when Leicester did it and won the league, you want to see people challenging, it gets boring, Liverpool Manchester City, they'll be the first two.
"We want to see people trying to topple over them and there might be another Leicester, there could be another Leicester around, who knows.
"Newcastle for me, if they can get top eight and a good cup run and build again, then top six, then suddenly you're getting that squad together, then top four, that's what they will be aiming for."