Rudi Garcia's men fell behind to Jonathan Biabiany's first-half opener but stormed back after the interval with goals from Alessandro Florenzi, Francesco Totti and Kevin Strootman's penalty.
It extends their winning start to the league season to three matches, while Parma have taken just one point from a possible nine.
Biabiany's goal six minutes before half-time was the first Roma have conceded in the Italian top flight this term, the Frenchman heading past Morgan De Sanctis to give the hosts hope of a first victory of the campaign.
But the visitors hit back in style after the break, with Florenzi turning home Miralem Pjanic's assist just two minutes into the second half.
They left it until the final 20 minutes to turn one point into three, though, with veteran midfielder Totti producing an excellent control and finish to put his side ahead.
Netherlands international Strootman netted his first for the club since signing from PSV when he converted a spot-kick following a foul on second-half substitute Gervinho.
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
