Roma and Napoli, the two biggest spenders in the close season, are the only teams to have won their opening three games. They are followed by Inter Milan, Juventus and Fiorentina, all on seven points.
Parma, under former Italy coach Roberto Donadoni, broke through six minutes before halftime when Frenchman Jonathan Biabany beat his marker to Mattia Cassani's cross and headed in at the near post.
Roma, who spent around 66 million euros ($88.1 million) in the transfer window with 16 million of that going on Strootman, levelled less than two minutes after the break when Miralem Pjanjic set up Alessandro Florenzi who blasted a right-foot shot past Antonio Mirante.
Strootman then cleverly set up Totti, 36, who stayed onside and placed his shot beyond Miranda in the 70th minute.
With five minutes left, Ivorian forward Gervinho was hauled down by Cassani in the area and Strootman rifled home the penalty for his first goal since his move from PSV Eindhoven.
Parma have taken only one point from their first three games.