A close-range effort from captain Grant Holt and a second-half brace by winger Andrew Surman saw the Canaries warm up for next weekend's Premier League opener at Wigan with a confident display in front of a 13,000 crowd.
Three of Paul Lambert's seven summer signings made the starting XI, with on-loan Tottenham defender Kyle Naughton a late replacement for full-back Russell Martin, who withdrew after the warm-up.
Parma, who finished 12th in Serie A last season, had former Liverpool defender Gabriel Paletta and one-time Chelsea striker Fabio Borini in their team, while veteran Argentina forward Hernan Crespo - who won the 2005-06 Premier League title at Stamford Bridge - sat on the bench.
Norwich, promoted back to the top flight for the first time in six years, should have been ahead on 15 minutes when Simeon Jackson capitalised on a slip from Parma captain Stefano Morrone and raced clear, but goalkeeper Antonio Mirante stood up to save.
The home side made the breakthrough on 34 minutes when Holt bundled in a deep cross from Naughton.
Canaries keeper John Ruddy was tested at the start of the second half by a 20-yard strike from Sebastian Giovinco, which he tipped over.
Norwich doubled their lead on 61 minutes when Holt broke into the left side of the Parma area before cutting the ball back to Andrew Surman, who slotted into the bottom corner.
Surman flicked in Anthony Pilkington's low cross at the near post with four minutes left to wrap things up.