The Magpies finished 11th in the top flight last season despite spending the majority of the campaign fighting relegation.
Some excellent transfer business in January saw United climb up the table under head coach Eddie Howe with the likes of Kieran Trippier, Bruno Guimaraes, Dan Burn, Matt Targett and Chris Wood all playing their part.
Targett has since made his loan move from Aston Villa permanent while Newcastle have also confirmed the signing of England international goalkeeper Nick Pope from Burnley for £10million.
Just over two weeks into the summer transfer window and the club has already spent around £25million on new signings. Looking ahead to next season, Parlour expects Newcastle to break into the top half.
“They could be a Man City, three or four years it took Man City to start bringing the players in, got the right manager,” he told talkSPORT.
“I think Eddie Howe will get an opportunity now. But I think a cup run is going to be important for Newcastle.
“Imagine them getting to Wembley and winning the FA Cup. That would be magnificent for the club.
“I reckon top eight for Newcastle next season, [and] a good cup run.
“Teams like Leicester did it, won the league. You want to see people challenging, it gets boring, doesn’t it? If you know Liverpool and Man City, I don’t know what order, but they will be the first two.
“We want to see people trying to topple [them] over and [Newcastle] might be another Leicester, who knows?”
Check out the latest Premier League transfer speculation below...
5. Seagulls consider Delap
Brighton are interested in Manchester City forward Liam Delap, who is likely to be available on loan this summer. (Daily Mail) | Getty Images
6. Gunners pull out of Raphinha race
Arsenal have withdrawn from the race to sign Leeds United winger Raphinha. (Sport) | Getty Images
7. Spurs lead race for double deal
Defensive duo Stefan de Vrij and Pau Torres are ‘in pole position’ to join Tottenham this summer. (Paul Brown) | Getty Images
8. Toon want Pepe
Newcastle United will reportedly now ‘make an offer’ to sign FC Porto winger Pepe in the ‘coming days’. The player is understood to have a £60m release clause. (Correio da Manha) | Getty Images