For the record, "pinch flat" means a flat that you get when you hit something hard while riding, like a rock, and pinch the tube between the rock and the rim. They don't result in 5-inch holes in your tube Pairs of little 1/4" holes, yes.

The problem could be a lot of things, including rim tape or rim strips that are getting between the tire and the bead and preventing the beads from getting under the rim's hook edge very well (assuming it's got hooks). I'd advise using cloth rim tape that lays in the bed of the rim, covers the spoke nipples, but is not so wide that it rides up into the bead area.

It also could be what HillRider said... your old rims just aren't ready for 80psi tires, or your tire's a little oversized, or maybe one rim is a little undersized (manufacturing tolerances).