The Parshendi could use caltrops on the likely Alethi approaches which would disable the bridgemen, or at least slow them considerably. They could use fire arrows on the bridges, and even if most of the bridges themselves didn't catch fire immediately, the arrows would still kill bridgemen and might set fire to other nearby bridgemen or even the bridges themselves causing them to collapse during the assault. They could set up a simple spearwall on the opposite edge of the chasm (the attacker side) to slow the assault giving them more time to shoot arrows. Lastly, and most simply, they could wait for the bridges to be placed and then push them into the chasm. They are mobile after all.
I think other people have addressed the issue of burning the bridges pretty thoroughly, but there is a point that's been touched on that I'd like to flesh out a bit. You ask why they don't use caltrops, or set up a spear wall. Besides possibly not having caltrops, the way they fight precludes them. The Parshendi do not fight on more than one plateau at a time. If you attack them, lose, and retreat, they let you go. They don't have to. They could jump chasms and harry you and bleed you white. But they don't. Similarly, they bring one one force to any given fight. Chasmfiend shows up, you get your army, they get their army, they meet you at the designated combat zone, and they fight you until one of you wins. That's it. We don't know exactly why they do this, but it clearly has to do with their beliefs about how combat is conducted.
I think the same thing explains why they don't use advanced troop formations. They fight in pairs, and each pair picks a guy, and takes him on. If they get beat, another pair might step up. And then another. But they don't do formations.
And you're right, BTW, that this isn't very smart. They know where these battles are going to take place. They don't need bridges to cross most chasms. They could send one force to a gemheart fight, and then have two more show up halfway through, attack the staging plateau, cut off the Alethi escape, and then overwhelm them.
But instead, they fight fair. One army of yours, one army of mine. Two armies of yours, two armies of mine. Each of your warriors gets to challenge a warpair, even if your warriors fight as a unit.
We don't know exactly what their code of honor is, but I think their way of conducting war is tied up in it, and that's why they fight the way they do.