The Rossoneri fought back from two goals down to level through Alessandro Matri and Matias Silvestre at the Estadio Tardini, but still left empty-handed as a last-gasp drive from Marco Parolo saw Parma claim a memorable 3-2 victory.
Allegri believes that a naive approach to the closing stages cost his side dearly, while insisting that Mario Balotelli must do far better after being hauled off at half-time.
He told Sky Sport Italia: "In the first half we created little and conceded twice on the counter-attack. After the break we did more and went close to a third goal, but then that free kick happened at the end.
"We were naive and conceded two goals just before the end of each half. We should've closed down and not allowed any more space, but instead we got it wrong.
"Balotelli had a few good moments, but overall it was not a good game. I needed a centre-forward who could really play that role and Balotelli wasn't doing it, although he has the characteristics. He'd also been booked, so I was afraid we'd go down to 10 men."
The Rossoneri sit ninth after today's defeat and will slip to 10th if Lazio earn a positive result against Cagliari later this evening.