i have dont this exact thing as it says in my signature.
i didnt do a DIY on it when i had it all apart but you basically want the red/yellow on the back of the foglight switch going to a constant source of power at the fusebox. clip the wire about 4-6 inches from the switch or just make sure you leave enough room to reattach it later if you need to. cap off the end that is not attached to the switch, the side that is attached to the switch will go to the constant power. next is at the relay itself. it is right near the keyless reciever under the dash, see picture. at the relay there are 2 wires comming from it that are red/white, one goes towards the under dash fusebox and one goes towards the foglight switch. cut the one going toward the under dash fusebox. you may have to split the loom open to see which goes which direction. once you know and have cut the wirecap off the end that goes towards the fusebox and the end that is still attached to the relay gets hooked to ground. this will allow the lights to come one regaurdless of headlights being on or not as well as allow fogs to be on with your high beams. the only thing to remember is that you will not have a light on reminder when you open you door as this is activated thru the parking light circuit.
the diagrams that are attached are the factory diagrams just for reference, not detailing the mod. and these are for the 06and prior ones, not the 07-up, they are different. mine was a bit different to do but this is what i did on my 04
the reason your prelude does that is because the parking light circuit is the feed for the foglights, not just the headlights AND there are 2 relays and several diodes that control the foglights in the 01 prelude, not just one relay and no diodes like the E. you can wire the + that is switched with the parking lights up to the input of the switch instead of going to constant in the fusebox so the parking light chime would come on, but then at least the parking lights would have to be on in order for the fogs to come on with the switch, and they would still stay on with the high beams if you wanted them to.