I'm an electrical engineer and I ran into this very problem. My problem was a faulty/intermittent brake switch. The part number is 191 945 515B, a two pin connector.
Removal is easy. It is in a socket where the brake pedal's steel meets it under the steering area where your feet are. Twist it to the right about 3/4 turn and pull it out.

What threw me off when I got the new one is the plunger seemed too long. What I discovered is the plastic plunger ratchets - meaning it will auto adjust to the brake travel.
You can actually push it in with a small amount of force to see how the plunger adjusts.

If you find yourself stranded, you can remove the shift handle (automatic transmission) with an Allen wrench on the left side of the rubber handle. The bottom plastic piece is 2 piece.
The part with the P, R, D, ... symbols pops off by prying up with your hands. The bottom plate comes off with 3 Phillips head screws. The solenoid that locks out the park shiftier is in the front
of the shift handle. When you start the car and depress teh brake, the solenoid will engage and pull down. When you release the brake, the solenoid will release. Baring any wiring issues, your problem is either the brake switch or the lockout solenoid. The brake switch is ~$12. The solenoid is ~$165.

I've heard the 4 pin brake switch is a little tricky because you have to install it correctly. If you have this type, Google that. There is a good forum that explains how to install these.

Try this
 

its called a shift lock solenoid common problem there was a recall for some cars in 2006 Anyways the part is expensive and can run anywhere from 190-400 $$$ to replace but check the connections first and clean with electric contact cleaner so much crap falls down into the shifter compartment including spilled drinks. Most of the time cleaning ht connectors will work. Or try jumping the connectors and see if it works its basically its a magnet that allows u to shift out of park. Its a safety device that all vehicles have. Or you can remove the shift cover and use a small screwdriver and push in the button to get you moving. Its not the brake light switch when u depress the brake pedal and u dont hear that magnet disengage u know thats the issue....

When its it's stuck in park - turn the key slowly til you hear a click while pulling into neutral. Just pull. You'll have to do it a few times. Once in neutral start the car then literally shake the shifter rough but not too rough. It'll go into drive and you're good to go. It's just a hack and some do it and some don't. There is no way to actually fix it. I have a 2003 vw golf.2.0 170k miles. Happened twice now. Takes like 10 minutes but it works.

Stuck in Park
 

Try replacing the brake light switch that is operated by the brake pedal. $10 part at Autozone same issue I had for months til I read a thread someplace else. Anyways when U push the brake pedal you should hear the solenoid disengage at the shifter if u dont hear anything try this simple fix. Havent had the problem since.

You are talking about a seperate problem similar issues though. One is a glitch in the system and one is an actual fixable problem. Great to bring up too! I was annoyed there isn't more info about these glitches that only happen in vw.. (Dad is a mechanic and he is in love with Golfs/gti - he suggested what you said, that wasn't my problem)

i have the same problem i can't trust it one day it works fine next time it's stuck.

Cannot do that either... The key cannot be removed from the ignition while the transmission is in Neutral.