Six minutes into the encounter at the Stadio Ennio Tardini, the Cote d'Ivoire winger broke the deadlock with his fifth league goal this season.
Bruno Alves doubled the lead for Parma in the 25th minute before giving a penalty away 11 minutes later.
The skipper held Babacar in the penalty area and the Senegal international stepped up to send Luigi Sepe the wrong way to half the deficit for Roberto De Zerbi's men.
Babacar who joined the Black and Greens from Fiorentina this summer has registered three goals in 12 league outings this term.
Parma held on to the lead to seal their sixth victory in the Italian top-flight this campaign.
Ghana's Kevin-Prince Boateng was in action for an hour while compatriot Alfred Duncan was introduced for Alessandro Matri with 12 minutes left.
Parma, now with 20 points from 13 matches, climbed to the sixth spot, two points above Sassuolo who are eighth in the Serie A log.