It was Roma's first for the season under new coach Luis Enrique.
Totti told his personal website: "Tonight against Parma came our first victory. I think it's a major achievement particularly in terms of morale.
"From this match we will draw a definite new certainty and trust in ourselves will be strengthened. The next game will be against Atalanta, which until now has scored three wins and a draw: without the penalty they would be practically first in the standings.
"For this we will face them with great respect but at the same time it will be essential to look for the goal with conviction and three points."