Fines. A parking ticket is essentially a notice that you have violated a city or municipal parking ordinance and that you owe the local government a set amount in fines. However, if you don't pay within a specified period of time, your original fee may double or even triple with late fees and collection fees.
Bench warrants. If you wait long enough on an unpaid parking ticket, a local court may issue a notice for you to appear in court to pay any balance you owe for parking tickets. Continue to ignore this notice and the court may issue a bench warrant for your arrest. That means the next time police see your car's license plate on the road, you may be going home in a squad car.
Car booting or towing. Even more likely than your arrest is the chance that your car will be "booted" or towed for unpaid parking tickets. Keep in mind, if your car is towed and impounded by police, they may search it without a warrant.
License suspension or revocation. Your privilege to drive in your home state can be suspended if you ignore unpaid parking tickets for long enough. (This is true even with out-of-state traffic tickets.) Driving with a suspended license is its own traffic violation, and it may lead to your license being revoked. Once revoked, you may lose your ability to legally drive for years before you can apply to reinstate your license.
Increased car insurance premiums. Insurance companies determine your premiums based on your level of projected risk, and your premiums can increase if you ignore parking tickets for long enough. The New York Daily News reports that while most auto insurers won't allow non-moving violations (e.g., parking tickets) to affect premiums, they will raise them for suspended licenses or failure to pay parking violations.
You can hope for parking ticket amnesty in your state, but the more realistic option is to resolve the ticket ASAP.