My son, sitting in his car parked in the parking lot of a hospital waiting for mom to return and was backed into by doctor working for the hospital.

You should contact an experienced personal injury attorney as quickly as possible.  Most personal injury attorneys represent plaintiffs in these types of cases on a contingency basis.  That means that you usually do not pay until you recover some kind of money in the form of a settlement or a judgment.  Once you recover, the attorney usually collects approximately 1/3 of the settlement/judgment.  There is a statute of limitations on personal injury actions, which means that if it is not filed within a certain amount of time from the date of the incident you lose any and all potential claims.  Since you do not say how long ago the accident took place, it is unclear if the statute of limitations is set to expire shortly.  However, it appears that you may have a valid claim since the potential defendants (the doctor and/or the hospital) both likely have insurance and the liability appears to be clear.  Therefore, it is your best interests to obtain legal representation.  Once you do so, the defendants' insurance companies should start taking this matter more seriously.

State Farm is your son's insurer? Then then have no choice but to accept responsibility. If you have collision, say, on your car, your company MUST pay (within the insurance limits, of course). Whether or not they can in turn collect from the doctor, the hospital, or either's insurer is their concern.

Take a look at your policy. If you have coverage for what you are claiming, inform your insurer that they have to pay you regardless of whether they can get hold of the doctor. If you want to motivate them to be more cooperative, inform them that if they don't pay, you will both bring the matter to your state's Division of Insurance and to the local media as a "shame on you" kind of story.

You cann also try to motivate the doctor to contract you by calling the hospital administration and informing them that one of their doctors backed into your son's car but refuses to cooperate with your son or his insurer; that leaves you no choice but to consider suing the hospital itself for the actions of one of its agents or employees which occured on hospital grounds. (They are responsible.) You will also consider going to local media with a story about how the hospital allows its doctors to hit-and-run. This will probably encourage them to pressure the doctor to call you--or possibly to see to payment themselves.