There is no law specific to "school parking lots". The law requires that each motorist keep a proper lookout and keep his vehicle under proper control. The idea that a person backing is "always at fault" is a myth. I could articulate dozens of scenarios where a person backing his vehicle would not be at fault for an accident.
Once your son began to back into the unobstructed roadway, any person coming along had a duty to see that he was backing and to wait until he had finished his maneuver to proceed. By trying to speed up and cut past your son, the other driver failed to yield the right of way to your son's vehicle which was lawfully occupying the roadway.
My comments are not intended to establish an attorney-client relationship, are not confidential, and are not intended to constitute legal advice. Proper legal advice can only be given by an attorney who agrees to represent you, who reviews the facts of your specific case, who does not have a conflict of interest preventing the representation, and who is licensed to practice in the appropriate jurisdiction where the legal issue may be filed or in the state where the law applies.
Sorry to hear about your son's accident. In general, there is a duty to yield to hazards in the road or path of traffic. Therefore your son would have an argument to make that the other car should have yielded while he was backing up. To get a true determination of the law in Ohio you should make an appointment and discuss the matter in detail with a car accident lawyer licensed in Ohio.
Best of Luck,
The Kane Varghese Law Firm
The bottom line is, the other student's father should be making a claim to your insurance policy, not to you. Let the insurance company deal with his "knowledge of the law"; that's part of the service you pay them for.
From your description it sounds like you would probably prevail, although I reserve any judgment of mine for cases where I personally investigate the facts. Certainly, however, your insurance company would hire an attorney to investigate this for you.