Jonathan Biabiany put the Ducali in the driving seat late in the first half. Second half efforts from Alessandro Florenzi, Francesco Totti and a Kevin Strootman penalty fired Roma to the top of Serie A ahead of next weekend's Derby della Capitale against Lazio.
Roma and Napoli remain the only two sides to have won their opening three league games. In contrast, Parma are still searching for their first victory of the season, with this being their second loss in three outings.
Parma missed injured pair Gabriel Paletta and Daniele Galloppa. Coach Roberto Donadoni rang the changes from the 3-1 loss at Udinese, bringing Walter Gargano and Marco Marchionni in for Gianni Munari and Jaime Valdes.
Roma's Michael Bradley and Mattia Destro were in the treatment room. After notching a debut goal against Hellas Verona, Adem Ljajic got the nod ahead of Gervinho to partner Florenzi and Totti in the Giallorossi attack.
Totti wasted a free-kick inside the first four minutes, firing weakly at Antonio Mirante from an inviting position.
At the other end, Antonio Cassano was afforded space and drove over from distance, while an outstretched Biabiany just missed making contact with a dangerous ball into the box.
Totti curled a tame 25 yard effort into Mirante's gloves, before a ball over the top found Maicon with space in the area. However, the Brazilian drilled his half volley into the side netting.
Biabiany spurned a great chance when he ballooned over in the area from a Cassano corner. The pace of the ball caught the Frenchman by surprise as he right footed it into Row Z. Totti and Amauri both saw weak efforts saved as the game started to open up.
Parma hit the front six minutes before the break. Biabiany made up for his earlier miss by heading Mattia Cassani's cross past Morgan De Sanctis at the near post. It was the first goal Roma have conceded this season.
Alessandro Lucarelli was lucky to stay on the field after a couple of rash tackles just before half time landed him in hot water. The irate defender got into the referee's face and was rightfully berated by Donadoni from the touchline.
Biabiany almost turned provider on the stroke of half time. The winger burst into the box, but Amauri was slow to react and couldn't connect with the Frenchman's pull back.
Rudi Garcia needed a response from his team and he got one within two minutes of the restart. A looping ball by Miralem Pjanic caught the Parma defence napping and found Florenzi in the area. The forward let it bounce, then smashed it past a static Mirante into the roof of the net.
Amauri unwittingly turned Biabiany's goal bound shot wide when he couldn't get out of the way in time.
Roma turned the game on its head and took the lead on 70 minutes. Massimo Gobbi played Totti onside, and the 36-year-old talisman took advantage of a high Parma backline, controlling Strootman's through ball, before coolly slotting home.
De Sanctis was left scrambling as a Stefano Okaka shot deflected just wide of the post. Marco Borriello had to clear an effort away from the goalmouth as Parma pressed for an equaliser in the final ten minutes.
The visitors put the game beyond the Crociati when they earned a penalty five minutes from time. Cassani tripped Gervinho on the byline and Strootman smashed the resulting spot kick high and down the middle, giving Mirante no chance.
Parma: Mirante; Cassani, Felipe, Lucarelli; Biabiany, (Sansone 77) Gargano, Marchionni, Parolo, Gobbi; (Mesbah 90) Cassano, Amauri (Okaka 69)
AS Roma: De Sanctis; Maicon, Benatia, Castan, Balzaretti; Pjanic, De Rossi, Strootman; Florenzi, (Marquinho 69) Totti, (Borriello 76) Ljajić, (Gervinho 54)
Goals: Biabiany 39 (P) Florenzi 47 (R) Totti 70 (R) Strootman pen 85 (R)