A run of six defeats and one draw in Parma’s previous seven matches was brought to an end thanks to the two spot-kicks which came late in each half.
Napoli, already through to the Europa League after winning the Italian Cup, remain seventh with three games left to play, as Parma move up to 10.
Skipper Lorenzo Insigne pulled Napoli level in the 54th minute following an Alberto Grassi handball.
But Kulusevski grabbed the win for the hosts three minutes from time after earning another penalty against Napoli defender Kalidou Koulibaly.
Inter Milan host Fiorentina later Wednesday looking to stall Juventus’ push for a ninth consecutive title.
Juve can seal the title with a win over 16th-placed Udinese on Thursday if Inter Milan, who are eight points behind the champions, fail to win at the San Siro.