Midfielder Dino Baggio, no relation to Juventus forward Roberto Baggio, lifted a lob past goalie Michelangelo Rampulla from the edge of the penalty area after receiving a perfect assist from Gianfranco Zola.
The ball slowly rolled into the net as Juventus defender Robert Jarni vainly rushed behind it in the attempt to clear.
It was the only goal in a tense match Wednesday night, featuring two teams which have been battling all season at the top of the Italian league and which will also play each other in the two-legged final of the Italian Cup in June.
La Juve had a reshuffled defense because of injuries and suspensions which sidelined Ciro Ferrara, Moreno Torricelli and German J--rgen Kohler.
The Turin team squandered several good chances, and Parma's goalie Luca Bucci made two sensational dives on powerful drives by Gianluca Vialli and Giancarlo Marocchi in the second half.
Colombian Faustino Asprilla led some dangerous counterattacks but was left isolated up front while his teammates defended the one-goal lead.
Wednesday's defeat at Tardini stadium, before 27,000 fans, was Juventus' first in UEFA Cup matches this season.
Juventus' French midfielder Didier Deschamps is one of five players who will miss the final leg because of second yellow card offenses.
Deschamps was punished for a rough tackle on defender Antonio Benarrivo, who had to leave the field with a strained ankle after eight minutes.
The second leg will be played at Milan's San Siro stadium, where the Turin club decided to play its games following a controversy with the owners of its home stadium, Delle Alpi.
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In the English premiership, Crystal Palace slid further towards relegation Wednesday when it lost 3-1 to a Southampton side inspired by a virtuoso Matt Le Tissier performance.
Palace's defeat on a night when fellow strugglers West Ham, Everton and Aston Villa all picked up a point, means Palace has to win at home to West Ham on Saturday -- and then avoid defeat in its final two matches at Leeds and Newcastle to prevent returning to the First Division just a year after coming up.
West Ham was scrambling after midfielder Martin Allen was sent off midway through the second half in a 0-0 home draw against Queen's Park Rangers. Nine other players were booked.
FA Cup finalists Everton twice led Chelsea but settled for a point in a 3-3 draw at Goodison Park, while Aston Villa scored for the first time in four matches in a 1-1 draw at Villa Park against Manchester City.
There was drama at St James' Park where Newcastle and Tottenham drew 3-3.
Both sides finished with 10 men after Newcastle's Czech goalkeeper Pavel Srnicek was sent off after 54 minutes and Tottenham's Scottish international center back Colin Calderwood was dismissed in the 65th minute.
While the relegation issue at the bottom remains confused, idle Middlesbrough was crowned First Division champion and gained promotion because Bolton could only draw 1-1 at Stoke.
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Paris St. Germain won the French League Cup final at home amid controversy Wednesday with a 2-0 victory during which Bastia was denied a goal on an incorrect offside call, two minutes before the end of the first half.
PSG could land a unique cup double when it plays Strasbourg in the French Cup final. (AP, Reuters)
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