Parma, winners in 1995, knocked out Rangers and will be joined by fellow Italians Bologna and Roma - who beat Leeds in the previous round - in the last eight.
Celta Vigo knocked Liverpool out of the competition, and their countrymen, Atletico Madrid, are also through. France claimed the remaining three berths.
Atletico Madrid, trailing 2-1 from the first leg, needed goals from Santi and Jose Mari Romero to kill off Real Sociedad after a double strike from Vladimir Jugovic, their Yugoslav international, had forced the game into extra-time. The capital club won 4-1 on the night and 5-3 on aggregate. Marseilles, Lyons and Bordeaux made this year the best yet for France in the Uefa Cup, which no French club has ever won.
A second-half goal from substitute Titi Camara, deputising for Fabrizio Ravanelli, the former Middlesbrough striker, enabled Marseilles to edge out their Mediterranean rivals Monaco 3-2 on aggregate at a seething Stade Velodrome, where Monaco have not won for 19 years.
Marseilles would have won by more than 3-2 on aggregate had Fabien Barthez, the France goalkeeper, not been in fine form for the visitors, who had their Bosnian international defender Muhamed Konjic sent off in the 38th minute for bringing down Christophe Dugarry.
Lyons won a pulsating encounter 4-3 at Bruges in Belgium with Alain Caveglia hitting a hat-trick to see his side through 5-3 overall. Bordeaux also went through, but only on away goals after a goalless draw against Grasshopper Zurich followed a 3-3 draw in Switzerland.
Roma, who squeezed past Leeds 1-0 in the last round, courted disaster before drawing 2-2 at FC Zurich to go through 3-2.
The Italians took a 13th-minute lead through Marco Delvecchio, but flirted dangerously with an unexpected exit as they survived a missed Swiss penalty and two second-half goals from the South African striker Shaun Bartlett. However, Roma's 22-year-old striker, Francesco Totti, came to the rescue with an injury-time free-kick.
Bologna could afford a 1-0 defeat at Spain's Real Betis having cantered through the first match 4-1. Betis centre-forward Oli Alvarez scored the only goal of the night.Reuse content