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...
1. Fulham lead Pellegrini race
Fulham are leading the race to sign 23-year-old Juventus defender Luca Pellegrini in the summer transfer window. (Calcio Mercato)
2. Branthwaite set for loan
Everton defender Jarrad Branthwaite is set to leave on loan next season, with Middlesbrough, Sheffield United, Huddersfield Town, Blackpool, Cardiff City, and Sunderland all interested. (Daily Mail)
3. Hammers make Danjuma contact
West Ham have approached Villarreal about signing winger Arnaut Danjuma. The wide man has also been linked with Liverpool. (The Guardian)
4. Magpies plot Osimhen swoop
Newcastle will make a ‘monster offer’ to sign rumoured Arsenal and Manchester United target Victor Osimhen from Napoli this summer. (Gazzetta dello Sport)