Auto selector sticks in park

Just a quick question to any one with an AL4 auto gearbox.
Randomly my shifter locks in Park. Most of the times its at start up, but occasionally when I park by the side of the road and leave the engine running but shift it into park, it also locks.
The solution seems to be to switch the engine off and re-start the car.
Another "quirk" is the selector "unlocks" in park when the engine is running. That is, I don't need to put my foot on the brake to unlock Park. On the surface it seems the solenoid is sticking. Yes, the dash indicates the correct selection (solid P, not flashing). So it's not the gear selector position switch on the gearbox. I still need to investigate the brake switch and the interlock solenoid relay (and the solenoid).
My question is: whenever you put your foot on the brake and the selector is in park and the engine is running, can YOU hear the interlock solenoid 'click'? As far as I know EVERYTIME the selector is put into park it should lock there and you need to have the engine running and your foot on the brake to unlock it?
Yes. I know there is an emergency unlock, but so far switching the engine off then on again does the trick.
While it looks like the lock out solenoid is sticking (open AND closed) why does switching the ignition off fix it? That's a rhetorical question by the way. LOL!

Tracked fault down.
Used LEXIA to monitor brake switch. It has two sets of contacts. 1 set is used by the BSI to turn the brake lights on and the other is used by the automatic gearbox to unlock the gear selector. The two sets should always be opposite, that is, if one set is open the other should be closed.
Brake contact 2 is used for the gearbox and was giving random readings. Sometimes open some times closed and sometime it worked correctly. These pictures show it working correctly.

I was going to say that lexia would be the way to investigate this. You beat me to it. Thank your for the answer and clear explanation of the problem.

Thanks Arfur.
To answer my own question, yes. IF the engine is running AND the shifter is in park AND you put your foot on the brake the shift lock solenoid will dis-engage every time. So if you tap and release the brake you will hear the solenoid engaging and dis-engaging but only in park.

When I first got the car the brake switch was loose and not attached to the bracket or wiring loom. When I re-installed it the brake lights were on all the time! The plunger on the top of the switch didn't move when the peddle was pressed, it was too short!? I then discovered the plunger was adjustable! It clicks inwards very easily but is very hard to pull out...but it will pull out. I would say the plunger was accidently pushed all the way in and then they abandoned it. But why did they pull the switch out in the first place? I wonder...LOL!

So I pulled the brake switch apart to see what was going on. Here is what I was confronted with. The actual contacts are molded into a carrier.
I used a small piece of fine 'wet and dry' sand paper wedged between each contact to clean them.
I put the contact carrier back into the white rear housing and plugged it back into the wiring loom. The contacts for the park lock solenoid still weren't working! A bit more cleaning with a fine screwdriver did the trick. Put the switch back together and re-installed it.
Presto! Working again...for now.