If the fuse blows when the grounds are connected, it has nothing to do with where you are connecting the grounds, it has to do with how you have the lights wired. It sounds like you have the bulb ground and one of the power wires reversed. The bulbs in your trailer have two filaments and, if they are 1157 or 2057 bulbs, two little bumps on the socket end. Each of those bumps go to one of the filaments, and both filaments are tied to the common metal base of the bulb. Somewhere it sounds like you have one of the wires for one of those "bumps" wired to ground. The parking lights work with the ground disconnected because the bulb is getting the ground through the second filament and the miswired connection that is currently shorted to ground.
If you are not willing to take the time with a continuity tester or a multimeter to figure out which bulb it is, buy a bunch of fuses, then pull all the bulbs, connect all of the grounds at all of the sockets, plug the trailer in with the lights on, then install bulbs one at a time until the fuse blows. Figure out the issue at that socket, then continue to the rest to see if it's only an issue with the one socket or multiple sockets.
-Rod