You probably have a faulty headlight switch. Replace it and that should fix it. Been there, done that.
You probably hit the parking light switch inadvertantly! The switch is on top of the steering wheel column and is a rocker switch. I think that pushing the switch away from you will turn the parking lights on and pushing it towards you will turn them off - it may be vice versa though! Parking lights are for use in some countries when you are parked on an unlit street so that other drivers can see your car.
possibly a bad headlight switch, or dimmer switch. Some later model buicks have a lighting module under the dash about in the middle. I think yours is too old to have that though.
You have what is called a back feed, where you have wires striped and touching. So start at the lights and start to trace the wires until you find the exsposed wires. This might take some time though.
One question I'd have is whether or not your tail lights are working. I had the lights on my dash go out (the shifter lights still worked though) and my tail lights and parking lights were also out. After much investigation, it ended up being that the turn signal switch had gone bad. In order to replace it, you need to remove the cover around the steering column - we couldn't get the lower part of the cover off and subsequently also had to pull the steering wheel, then remove the turn signal switch (which also controls the headlights, parking lights, tail lights, etc), pop in a new one, and enjoy a life of dash lights and tail lights once again.
If you are referring to the Daytime Running Lights (DRL's), no. When you are not moving you can turn them out by activating the parking brake, but there is no way to turn them off without actually pulling out the fuse that controls the circuit. Careful though, I don't know what effect this might have when you actually want to turn your lights on at night.
I would suspect a bad headlight and / or turnsignal switch assembly. I just ran across this problem recently with an 89 Aerostar with an Astrostart remote starter. When using the remote to lock the vehicle, the parking lights flash on and off. Something with the Astrostart was keeping the parking lights on until I shut the power off to the remote starter and from that point on the parking lights shut off fine. I don't know though what in the Astrostart is causing it. I don't know anything about remote starters. The owner is happy that his battery no longer goes dead but now he has to get help with the remote starter.
Look at the wiring bundle that feeds the headlights. The wires will eventually break because of the constant up & down flexing of the wire bundle (motorized hidden headlamps). The problem will likely be found within inches of where the bundle is secured to the inside of the popup headlight carrier (where the flexing would be concentrated).
The headlight switch may be bad. Try replacing. Its the Headlight switch. The switch has a built in Circuit Breaker that self resets. It has become bad and will eventually give it up.
Usually it is a warning that either you have left your keys in the ignition, or you have left your headlights, interior, or parking lights on. Sometimes, though, the little switch in the ignition that senses the presence of the key, will stick. Reinserting the key and jiggling it around a bit while removing it will often fix it.
They could be burned out.
The headlight switch that has the dash light dimmer knob. The dimmer switch went bad and you can only fix it by getting another headlight switch on most cars. Some though have a separate dimmer switch.
Is the circuit grounding out? Bare, broken, corroded, rubbed though wire?