I will TRY and do this one quickly -
The very second you stop your vehicle on the premises of a parking facility (private or otherwise) you have bound a contract between you and the entity that either a) owns the facility or b) is legally entitled to operate a commercial enterprise (i.e. parking lot) on that premises.
The contract is formed by the ticket that you take out of the machine. The terms of the contract are either written on the rear of the ticket or in another accessible location to which reference is made on the rear of the ticket.
Now you are bound to a contractual agreement between you and the car park. One of the fundamental conditions of the contract is in regard to the amount of time in which your vehicle may remain stationary on the premises and the compensation you are entitled to provide the car park owner with (i.e. $x per hour). The terms will also list the quantity of any fine that may be incurred for parking to long.
So if you have incurred a fine from a private car park then it is going to be deemed to be a commercial charge. By not paying that charge you are doing the equivalent of not paying a tax invoice.
The car park must provide you with an opportunity to pay, and then if the invoice (fine) remains unpaid for a set period of time, they can issue a Letter of Demand with terms of payment. After 14 days of non-payment the car park business can issue a Statement of Liquidated Claim through the local court.
After the statement of claim has been served on you and a hearing has been scheduled, you will be required to appear in a local court. If you are unable to tender a sufficient reason as to why the amount has not been paid then a judgement will be passed and you will need to pay the outstanding amount plus the car parks legal expense.
So that is the legal right of the car park operator to recover the outstanding expense, however........ would they pursue it that far.... doubt it!
90% of the time these charges are what they call "opportunistic revenue", the car park operators are trading on the fact that their fine is low enough and the chance of further charges is enough for you to just pay the thing.
The chance that they will pursue you through the courts to recover such a small amount is fairly remote.
Also, don't make the mistake of confusing this with a fine. A "Fine" in the true essence of the word can only be enforced by a local government entity for the violation of a statute or by-law. They certainly don't have the ability to "prosecute" you in any way. What you have incurred is a commercial expense as outlined in a legal contract and it will only ever be a commercial arrangement in regards to compensation for a service.
So once again, I really doubt that they will pursue you legally. They may try to scare you into paying, but I doubt they would go legal. However if they do make the decision to pursue you then you should understand that you could be up for greater expense and if a judgement is passed it will be listed on your personal credit file as a judgement.
P.S. This is not advice.