In an action-packed Serie A clash which included two red cards as well as nine goals, Milan came out on top thanks to Jeremy Menez's brace.
Milan led 1-0, 3-1, 4-2 and 5-3 but continually had to fight off Parma's responses at the Stadio Ennio Tardini.
Donadoni said there was no difference in class between the two teams, rather his side's errors proved costly.
"We were naive on two or three occasions and conceded goals more due to our errors than the quality of our opponents," he told Sky Sport Italia.
"Milan did not outclass us tonight and this game must make us even prouder of what we can achieve when we put our minds to it.
"We only made a few individual errors, as did the referee."
Giacomo Bonaventura put Milan ahead with a clever turn and finish before Antonio Cassano headed in an equaliser.
Milan were 3-1 up at half-time thanks to a Keisuke Honda header and Menez's clinical finish from the penalty spot.
Felipe pulled a goal back for Parma, who looked on track to take something out of the clash when Daniele Bonera was sent off for Milan.
Nigel de Jong and Alessandro Lucarelli traded goals before Felipe saw red, and Milan won after Menez's second goal - despite Mattia De Sciglio's unusual own goal.
Donadoni said the red card to Felipe cost his side and was a wrong decision.
"I would like for certain incidents to be commented on, as they can change the course of a game," he said.
"It looked as if Felipe did absolutely nothing and when fighting back for the result this red card undoubtedly penalised us.
"If we also look at Felipe's first yellow card, I honestly don't realise how a referee can make such a big mistake. It's incredible, not just for us, but for anyone in this sport. To see that many errors in one game really gets you angry."
