Both Crespo's goals were set up for him by the defender Paolo Vanoli in the 37th and 66th minutes, while Chiesa scored with a long-range shot in the 43rd and rounded off a counter-attack by converting a pass from Crespo in the 59th.
Crespo's compatriot, Juan Veron struck a brilliant free-kick just after half-time, and Abel Balbo made it four for the Argentinian contingent with a late penalty.
Bordeaux, who lead the French league, arrived at Tardini stadium intent on protecting the 2-1 advantage they gained in the first leg two weeks ago, but their defensive strategy backfired.
Victory kept alive Parma's quest for a rare treble; they are in the Italian Cup final and are tied for second place in Serie A. But the 1995 Uefa Cup-winners will have to play their semi-final first leg without defensive stalwart, Fabio Cannavaro, who collected his yellow card of the tournament after being booked for handball in the 54th minute.
Otherwise, everything went Parma's way. In the 37th minute, Diego Fuser's pass found Vanoli on the left, he drew the goalkeeper, Ulrich Rame, and slid the ball inside for Crespo's first. Then Vanoli's dummy run left space for Chiesa to collect Dino Baggio's long pass for the second.
Just after half-time, Veron curled in his free-kick to make it 3-0 and Chiesa took Crespo's through ball to fire in the fourth. Then Crespo scored from Vanoli's nod-down of a 40-yard pass by the French World Cup winner, Lilian Thuram.
Balbo got the last goal, with two minutes to go, after the Bordeaux defender Nisa Saveljic, standing near the penalty spot, inexplicably grabbed the ball out of the air with both hands.Reuse content