parking brake switch is probably dirty and/or needs replaced. It's a simple plunger type switch. And when the indicator light is not coming on, the contact is either bad or plunger is not deploying.
Cost: ~ $5
p/n - 996-613-112-01
In a pinch, try pulling up and releasing the handbrake lever either gently or rapidly multiple times until the light on the dash comes on. It usually works
R&R will take you 25-30 min.
All you need to do is pull up the center console. You don't need to completely remove it, just get the rear end up about 12". Then unplug the 2 wire lead, release bad switch from metal frame of the hand brake base and put the new one in place. IIRC, you can just gently pry the old switch out (know you'll most likely break the old plastic tabs off, but the new switch comes w/ replacements). Plug the wire leads back in.
Normally I'd say just try and clean it first but since the part is $5, I suggest just replacing - especially since you have an '01...
