Afriyie Acquah of Parma cancelled out Gervinho's opener in the 15th minute before Francesco Totti scored his 234th league goal to put Roma ahead one minute later.
Miralem Pjanic celebrated his 24th birthday by extending Roma's advantage four minutes into the second half before substitute Rodrigo Taddei scored the hosts' fourth.
Parma's Jonathan Biabiany knocked in his team's second in the final minute.
The outcome is Roma's fifth straight league win and has left the Giallorossi in second place, nine points clear of third-place Napoli. Parma are sixth in the standings and now winless in four games.
The contest at the Stadio Olimpico was originally scheduled to be played in February but was postponed eight minutes into the contest due to a waterlogged pitch.
Roma struck first in the 12th minute after Destro's right-footed shot hit the post, where the awaiting Gervinho pounced to tap in the rebound.
Gervinho then had a goal disallowed for offside.
Parma equalised soon after when Massimo Gobbi's cross found Acquah in the heart of the area and his right-footed shot beat Roma goalkeeper Morgan De Sanctis.
However, the hosts went back in front almost immediately when Pjanic fed Totti in the box and the veteran captain struck the ball into the top left corner of goal and out of Parma goalkeeper Antonio Mirante's reach.
Parma almost levelled again on the half-hour mark but Carvalho Amauri's goal-bound attempt from 30 yards was kept out by De Sanctis.
Roma were unlucky not to extend their lead before half-time with Destro's powerful header from Maicon's cross struck the crossbar.
Roma picked up where they left off after the re-start and made it 3-1 early in the second half.
Gervinho's shot took a deflection off a Parma defender and fell into the path of Pjanic, allowing the Bosnian playmaker to drill a right-footed shot past Mirante.
The visitors missed a good chance to reduce the arrears before the hour mark when Gobbi's header from Mattia Cassani's cross went just wide.
Roma continued to attack and were again rewarded in the 82nd minute when Taddei got his head to substitute Alessandro Florenzi's cross and nodded past Mirante.
Parma scored seconds from time when Amauri spotted Biabiany on the edge of the area and the Frenchman's right-footed shot beat De Sanctis.