Antonio Di Natale made it eight Serie A goals for the season after nine minutes, with home goalkeeper Zeljko Brkic ensuring Parma would not get anywhere near an equaliser before the break.
But Marco Marchionni notched for the first time since returning to the Ducale club, only for Roberto Pereyra to nudge Udinese ahead once again.
Donadoni was desperate to avoid a second consecutive away defeat, though, and his decision to throw Palladino on as a second-half substitute paid dividends.
Di Natale registered his 161st Serie A goal in typically consummate fashion, controlling Maurizio Domizzi's ball forward to coolly beat Antonio Mirante and put Parma on the back foot.
The veteran striker thought it was goal number 162 three minutes later, but the offside flag ruled otherwise, and soon Parma bared their teeth.
Brkic had to produce a save of real quality as Amauri arrived in space to head at goal following fine build-up play from the visiting team.
Di Natale soon weaved down the right, finding Maicosuel in the area, but Gabriel Paletta got the block in at a crucial moment.
Giovanni Pasquale came close to doubling Udinese's advantage before the break but he could not quite convert Emmanuel Badu's cute pullback.
Indeed, Parma should have been level at the interval if not for Marco Parolo's faulty radar.
Brkic steamed out of his area to head clear a loose ball, but he could only find the Parma midfielder, who lofted the ball wide of a gaping net.
The disappointment did not last long among the Crociati ranks as Marchionni levelled direct from the restart.
The tricky winger scored his first Parma goal since January 2006 by dribbling into the area from kick-off, shaping himself to beat Brkic low to his right.
Pereyra was happy to play the role of party pooper, though, finding himself well-placed to make good on Dusan Basta's parried effort in the 50th minute.
Di Natale and Maicosuel tried to catch Parma cold but their swift interplay was foiled by Mirante, before substitute Dorlan Pabon missed the chance to drag the Crociati level eight minutes from time.
The Colombian forward came face to face with Brkic, but the keeper guessed right and the ball was cleared to safety, for the time being.
The visitors kept pushing for a final goal and they found it one minute from time.
Parolo unlocked the Zebrette backline and substitute Palladino was waiting to lift a sublime lob over Brkic and into the back of the net.
It was Parma's 1,000th Serie A goal, but Yohan Benalouane could not make it 1,001 as he headed wide of the target from seven yards.