First come, first served works fine if you all start at the same time but I found you're often at the mercy of timetabling.
If you have high fees, you’d be just transferring the scare resource to the rich. And the people who can’t afford it are forced to figure something less convenient out.
It also means you can't plan effectively. If half the time, you show up to class and can't park anywhere, how can you make it to class on time consistently?
If your answer is "well, maybe you should take public transportation instead of driving", how is that any better for students than charging for parking?
If you charge for parking, you are at least giving students a choice and the ability to make an informed decision: "is it better to drive and pay $x to park, or save money and take public transportation?"
If you have free parking that is often full, your choice is "take public transportation or have a x% chance of not being able to park and missing class"