1994 Diamante that I inherited from my father-in-law needs a lot of TLC. My biggest problem that is driving me nuts is this one. The parking lights stay on all the time! The headlights, etc.. work perfectly but when the headlight switch is turned all the way off, the parking lights and dash lights remain on. I replaced the Headlight Switch Assembly on the steering wheel this morning and it didn't resolve the problem.

What else could it be?

 

Check the relays under the hood in the fuse/relay compartment. It could be a stuck relay. If you have a owners manual it will point you directly to the relay.
If not try pulling one at a time and see if the lights turn off.

 

20 bucks says you have an aftermarket radio installed, dont you?