Dealerships lie. They told you that dad would be a cosigner, but the likely only way the lease could be approved is by him being the primary. They likely did a clever job of concealing that fact, and if your dad signed the paper work, both you and he are co-owners. Any derogatory mark on your credit would equally be on his credit. You two share an equal fate in the outcome of this transaction, so it is best if you two can work together.
When we are adults and make mistakes it costs us money, that is how we are "punished". This transaction contains a number of mistakes, by the both of you, so I would be prepared to pay to make them go away.
You should probably pay for these. You may not have noticed the illegal parking signs or they may be cleverly hidden to trick unsuspecting tourists. It is far easier to give you a ticket via license plate then actually put one on your windshield. Plus, you will likely be a repeat offender if you don't have a physical ticket. You can fight them, but perhaps the best you can hope for is a reduction in fine.
Dad off the Lease
This will require a separate car transaction. Since it is a lease you will have to obtain the pay off amount and either get a loan on your own for that amount, or a new lease. How likely are you to qualify for a loan/lease on your own? How much did you put down on the original lease?
Unless you put a large amount down, you will likely be unable to obtain a loan or lease for the pay off value of the car. This means you will have to come up with cash to make this transaction happen.
Co-signing is something you should never do for someone, nor should you ask someone who you care about to do for you. This is especially true for such as a diminishing asset, like a car. These kinds of transactions lead to all kinds of "tales of woe".
Leasing a Car
Leasing a car is the most expensive way to drive a car. One is probably better off renting a car from Hertz or the like on a continuing basis. The ramifications of this transaction can echo into your financial future for many years. Some people never get out of the car payment cycle. It is best never to begin, but you already started.
The best thing you can do is pay cash for cars and save for your next one. Some people, who have actual cash, choose to invest the money if they can obtain a low interest loan. Fine, but that is a very different situation on one who borrows/leases a car with little in the way of assets.
Other than the parking tickets all of this pain and suffering would have been avoided if you paid cash for your car. Your relationship with your father would not be suffering. Sure the banks would have less money, but do you really want your paycheck to go to bankers?