To literally do what you desire, you could run a wire from the parking light circuit (non-dimming brown wire at the headlight switch or driver's side kick panel as I recall) to the power terminal on the fog light switch. Insulate the old power feed that you're now no longer using.
I would go back over your wiring connections (done in the retrofit) and find the bad connection. Check for KOEO power going into the switch and then leaving it. All you're doing in the mod is switching power sources.