A dreadful stoppage-time miss by forward Milos Krasic cost Juventus, who started the weekend as joint leaders with AC Milan, the chance to win after they squandered an early lead.
Krasic, out of favor under coach Antonio Conte this season, was given a rare chance in the team and failed to take it as he sent a weak effort wide of the mark when he was presented with a free shot inside the penalty area.
Mirko Vucinic had given unbeaten Juventus a seventh-minute lead only for Andrea Cossu to equalize three minutes after the break with a 25-meter shot that swerved away from Gianluigi Buffon.
Juventus have 38 points from 18 games, one more than AC Milan.
Donadoni, sacked by Cagliari before the start of the season, got off on the right foot at Parma thanks to goals from Jonathan Biabiany, Francesco Valiani and Sebastian Giovinco, leaving them 12th with 22 points.
Donadoni replaced Franco Colomba, sacked by the Sardinians after last week's 5-0 defeat at Inter.
Third-placed Udinese lost ground in a 3-2 defeat at Genoa where they were missing three regular players — Kwadwo Asamoah (Ghana), Emmanuel Agyemang-Badu (Ghana) and Mehdi Benatia (Morocco) — who have been picked to play for their countries in the African Nations Cup, which starts on Saturday.
Although Damiano Ferronetti gave Udinese a halftime lead, they were stung by three goals in a 21-minute spell from Andreas Granqvist, Bosko Jankovic and Rodrigo Palacio.
Striker Antonio Di Natale, 34, pulled one back from a penalty, his 13th goal of the season.
Udinese have 35 points, two more than fourth-placed Lazio, who narrowed the gap with a 2-0 win over 10-man Atalanta.
Hernanes converted a first-half penalty and Miroslav Klose added the other in stoppage time, both celebrating with acrobatic somersaults, as Lazio ended a three-match winless run and bounced back from last week's 4-0 defeat at Siena.
Romania forward Adrian Mutu scored twice, including a penalty, to give Cesena a 3-1 win over Novara, leaving them joint level at the bottom with Lecce who won 1-0 at Fiorentina.
