The Bianconeri were beaten to the group stages of the Europa League by Slovan Liberec and lost their Serie A opener to Lazio but goals from Emmanuel Agyemang-Badu and Thomas Heurteaux put them on the right track.
Former Inter Milan striker Antonio Cassano opened his account for winless Parma with time running out but Luis Muriel would guarantee all three points with a penalty two minutes from time.
After some early teasing in the midfield Udinese took control of the game when Badu opened the scoring in the 11th minute.
The Ghanaian midfielder was in the right place to tuck home after Mattia Cassani failed to deal with a probing pass from Antonio Di Natale.
At the other end, Amauri won a tussle with Naldo to get his head to Marco Parolo's cross and slam it against the bar.
Di Natale turned up the heat with a couple of solid chances before the half-hour mark, first giving Antonio Mirante a scare with a searing right-footed effort before clattering a post when put through by Andrea Lazzarri.
Parma needed a confidence boost before the break but they would head down the tunnel frustrated as Cassano clipped Jaime Valdes' pinpoint cross wide of the mark.
The Gialloblu were bright following the interval but Valdes' effort from the edge of the box sailed wide before Parolo's speculative punt skipped harmlessly wide of Ivan Kelava's goal.
Udinese waited patiently for a chance to break again and they finally did so in the 65th minute, with the rapid Muriel escaping his marker to force a desperate punch from Mirante.
Heurtaux then found himself alone in front of Mirante to convert a deceptive short free-kick from Maicosuel.
The Frenchman might have produced the third goal in the 78th minute but a wonder save from Mirante kept the hosts waiting.
Parma fans had been waiting to see a first goal from summer signing Cassano and they were sated when he rose highest to nod home Aleandro Rosi's cross with eight minutes left.
It was a false dawn for Roberto Donadoni's men, however, as Udinese were awarded a penalty two minutes from time.
Lazzarri went down in the box under the attentions of Pedro Mendes and Muriel made no mistake from the spot.