Striker Pablo Osvaldo's second-half header was enough for ambitious Roma who still had to withstand a late pounding from the hosts.
Juventus, who drew 1-1 at Catania, stayed unbeaten and share the Serie A lead with Udinese, who are also undefeated after a goalless stalemate at Cagliari. The two sides have eight points from four games.
Promoted Atalanta remain the only other unbeaten team after a 2-1 win over Novara with goals from Matias Schelotto and Luca Cigarini.
They have the best playing record with three wins and a draw but have only four points after having six deducted before the start of the season in a Serie B match-fixing scandal.
Genoa's unbeaten record ended when they lost 2-1 at Chievo, who also beat Napoli in midweek.
Striker Rodrigo Palacio, recalled by Argentina last week after a three-year absence, put Genoa ahead just after half-time with his fourth goal in as many games.
Sergio Pellisier levelled for Chievo in the 74th minute and Davide Moscardelli scored a stoppage-time winner.
Siena beat Lecce 3-0 helped by a brace from Emanuele Calaio after Mattia Destro opened the scoring, while Lazio and Palermo played out a goalless draw.
ATTACKING FOOTBALL
Luis Enrique has promised to turn Roma into an exciting, attacking outfit but Europa League elimination at the hands of Slovan Bratislava followed by failure to win their first three Serie A games quickly put him under pressure.
The only excitement in a dismal first half came when Francesco Totti's shot was pushed onto the post by Antonio Mirante.
The second half was a different story as Osvaldo headed in from Aleandro Rosi's cross five minutes after the re-start.
Roma, who have five points, controlled much of the play but lost their grip towards the end.
Parma poured forward but a combination of dreadful finishing and some desperate defending from Nicolas Burdisso and Gabriel Heinze kept Roma's goal intact.
Juventus fell behind after 21 minutes at Catania, Alejandro Gomez supplying a low cross which was turned into the net by his fellow Argentine Gonzalo Bergessio.
Serbian striker Milos Krasic equalised four minutes after the re-start with a right-foot shot which was badly misjudged by goalkeeper Mariano Andujar, another member of the Argentine contingent in the Sicilian side.
Juve had a let-off late in the game when a mistake by Giorgio Chiellini set up a chance for David Suazo but the Honduran was foiled by Gianluigi Buffon
