I'm going to take a photo out of the window. It's truly shocking.
We have 68 parking spaces and over 200 employees.... the industrial park is building another new building next to us (only spare place left). I would hope and pray they add in a multi story car park but I know they won't. I have no idea why the owners selected this place when the parking is so inadequate. Parking gets absolutely ATROCIOUS as you work your way further into the park.
Yet immediately behind our car park is overflow parking for J&J and they don't use it. But if we park there, we are asked to move our vehicles immediately. It's insane. Luckily I have an assigned spot. 
The keys at the desk is an idea. But we really need to reduce the amount of cars on site if at all possible. No idea how it's possible though. I mean, if someone lived near me, they aren't going to want to go with me to drop off kid #1 at location #1 and then kid #2 at location #2.... not to mention the arrival/leaving times then being reliant upon someone else...
Gosh sounds like my work. Our car park we used to rent got bought out and taken away, so we now have about sixty spaces in front of the office for at least 200 people. Some are reserved for visitors, some for people with mobility needs. Which limits it even more. Somebody built a brand new car park in between our office and another but we cannot use it. If we park in the company we share offices with spaces they get so mad and threaten bringing in ticketing us.
We do have street parking but sigh. I’ve taken to parking about half mile away and walking in.
Our office facilities team walk around mid morning and you HAVE to register your car with them. If you don’t they will send mass emails with your registration number until you do.
We do have a bank of spots that can only be used for those car sharing. You must register for the spot and can only park in the spots if their free after ten pm. Our work also offers a cycle to work scheme ( not ideal in winter).
Somehow or another we make it work, but I feel the pain your office does.