The visitors took just eight minutes to break the deadlock, as Erik Lamela shot home from the right side of the box.
The hosts eventually equalised in the 34th minute when Ishak Belfodil found the target from inside the Roma box.
Three minutes later Marco Parolo turned in Belfodil's cross to make it 2-1 to Parma.
Midway through the second half Parma went 3-1 up at home, thanks to a Cristian Zaccardo shot from close range.
However, Totti brought Roma back into the match in the 71st minute with a left-footed finish from the centre of the Parma box.
Belfodil's dismissal with five minutes left on the clock gave the visitors some extra hope, but ultimately made little difference.