The car renter is responsible for parking / traffic violations.
Why you should pay rental car parking / traffic violations right away:
Parking tickets are tied to the car, not the driver, so there is zero chance the rental car company won't ever find out about it. This is why it's so important to pay the fine quickly. The faster you pay, the less likely it is that the rental car company will be notified.
Somewhere on the parking ticket there will be a section that details multiple ways to pay the fine. Typically, options include paying by phone and paying online. If at all possible, choose one of these quick fixes before you return your rental car. Be sure to keep a confirmation of the payment in case you need to contest being charged a second time by the rental car company.
If you got a ticket while driving a rental car in another country, the authorities can also track you down. While there are many online stories of individuals ignoring parking tickets received outside the US and never seeing a bill, don't try it. The parking ticket will eventually work its way back to the rental car company, which will simply charge you for the ticket and administrative fee down the road.
Unpaid fines and rental car administration fee balances will eventually be sent to a collections agency. In addition, you might end up on the rental car company's dreaded Do Not Rent list. It's best to pay your fine and move on.
What if you are unable to pay the tickets prior to returning the car?
The 'violations' team of the car rental company settles the tickets and charges your credit card along with an administration fee.