| The result means that for the first time since the Heineken Cup's conception Wales will not have their maximum quota of four teams in Europe's premier club competition.
The visitors opened the scoring in the opening minutes with a long-range penalty from Francesco Mazzariol but the Dragons responded with two tries - both from prop Adam Black.
On both occasions the Dragons opted to kick to the corner for the lineout and the catch and drive brought them due reward.
The Welsh side failed to nail down their advantage and Parma, down to 14 men following the sin-binning of prop Fabio Staibano, rallied when flanker Liviu Pascu exploited a gap in the Dragons' defence to race away and score.
The scores were tied once more when Mazzariol added the conversion and he later added a superb penalty to give his side the lead.
The Dragons edged ahead again when Nathan Brew linked with full-back Gareth Wyatt to put winger Gareth Chapman in at the corner to give the Welsh team a 15-13 half-time lead.
The lead changed hands again after the break when Mazzariol added another penalty, having missed two other attempts either side of half-time.
Black had a third try ruled out by the television match official before centre Ezio Galon landed a huge drop goal to give his side some breathing space.
The Dragons desperately tried to rescue the game but their hopes were finally ended when Paul Emerick exploited another gap to score.
As a result the Dragons will now play in next season's European Challenge Cup.
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