But Ebbw Vale crashed out of the tournament as they went down 63-17 at Pau.
The Castletown Men progress through by eight points thanks to their 41-28 victory at Virginia Park last weekend.
Caerphilly started the stronger and took a 10-9 lead into the break, but three second-half yellow cards proved costly.
Taumaholo Taufahema scored the only try of the game after superb forward play had allowed Sam Cox to break through and feed the Tongan international who touched down.
Justin Thomas added the conversion and a penalty goal but the Italians responded with three penalty goals from full-back Corrado Pilat to trail by one point in the game and 14 points overall.
In a scrappy second-half Mark Ring's side lost their composure and discipline as the Italians started to come back into the game.
Early in the half Faeo Vunipoloa was sent to the sin-bin, and he was followed by Paul Jones and Nic Rouse.
But despite intense Italian pressure, Parma could only manage two second half penalties.
Meanwhile in Pau it was Welsh exile Martin Pearson who did the damage to Vale's Shield hopes.
Pearson scored a try and kicked goals from all over the park as Pau overturned a 26-point first-leg deficit in stunning fashion.
The Steelmen's 29-3 victory at Rodney Parade last weekend was quickly forgotten as Pau welcomed back several star players and ran riot.
Damien Traille picked up a brace of tries and international No 8 Imanol Harinordoquy, who missed last weekend's match, also got on the scoresheet.