DO THEY ADMIT THEY HIT YOUR CAR? If the driver admits he hit the car, the manager can not say the otherwise.
Do not let them confuse you. Stick to the question: did the drive hit your car and cause the damage. Make this certain before talking about settlement.
If they do not admit, talk to your neighbor. Ask him to be the witness in the court. And tell the manager that you have the witness. You will sue them in the court.
If they admit, make sure your car was parked legally. Then they have 100% liability. It is not you who said the repair cost is $xxx or $yyy. It is the body shop. So you are not blackmail them.
Remember:
1. You can choose your body shop. They can not specify which body shop you should use. Their insurance company usually will ask you to estimate the cost at one of the shops they know, otherwise they will verify the damage/cost to make sure they are not paying too much. Anyway for the case like this, the manager will likely try settle with you w/o involving their insurance company.
2. You do not have to use the money to actually repair the car. You can put the money in your pocket, fix the damage yourself, or whatever.
3. For such simple case, you do not need a lawyer. If they do not admit it, report to the police first. Then file the complaint at the court.
4. The neighbor is the key. The driver is also important. Get hold on to them if you can. If the neighbor is willing to testify, just tell them you have the witness and you are going to sue them, they will yield and pay you.
5. Keep your thinking clear. First, your car was legally parked and they hit you. Then proceed to the settlement. Don't mix things up. And don't let them mix things up.