Parking fine

Hello, I just found out I got a parking fine and I have a couple of questions to people living in Tallinn and knowing local rules.

As I understand the I got the fine because I left a car parked on a grass. I payed for using this specific parking lot, but unfortunately there were no free spots so I parked at the grass right by the taken parking spot (yes, that’s not how I should use this parking but I’m sure I didn’t couse any trouble to people who may have wanted to drive/walk by).
Anyway I understand the fine was given fairly.

Now the question is if that is a legit parking ticket? Is that given by an organization who has the right to make me pay? In my home country there are often situations when for example supermarkets give this kind of fines without any law standing behind them hoping that people will get scared and pay so I want to know if it isn’t a similar thing here. (then you can freely put the ticket in trash without paying anything)

The ticket also says that if I don’t pay the fine they may search for the vehicle and driver info in a vehicle database to reach me. Can they do that as the car is registered in another EU state?

Now supposing that the ticket is legit and they can indeed find me in their databases: can I somehow request for a waiver as this is my first offence of this kind, it didn’t cause any trouble to anybody and the car was parked safely?

I got quite nervous about this fine as I’m new in Estonia and don’t really know how these things function here.

Thank you for all the answers