> The family has proof of residence (which is its own absurdity we won't discuss), and this third party can arbitrarily override that based on a black box argument.

Doesn't the family have a very straightforward libel claim against the third party? That the car was parked elsewhere may be true. "Although you are the owner on record of a house in our district boundaries, your license plate recognition shows that is not the place where you reside" is a statement the family can disprove in court (to a civil standard) and demonstrate has financially damaged them ("her daughter is currently attending a private school 45 minutes away from her home"). If that statement came from the third party (rather than the school district misinterpreting the raw data themselves), the family will win. The straightforward financial damages (let alone anything pain / suffering / punitive damages) likely exceed the company's payment from the school district ("a total of $41,904 for a 36-month-long contract"). It wouldn't take many of these claims before the company becomes insolvent, and good riddance.

I'd also expect them to win a lawsuit against the school district for falsely denying the basic right of education. Perhaps the individual school administrator also for libel. With any luck, a total legal bloodbath that warns any other school districts away from this conduct.

That depends if the third party makes the claim of non-residence and how they make it, and if they disclaim warranty and reliance. I can show you a site with some graphs and data of who is parked where and when and how often; I doubt they're directly saying, "This person definitely doesn't live at this residence, so deny her child entry."