A few key points:
The only way you can recover anything is with a lawsuit or the threat of a lawsuit.
There are special procedural and substantive hurdles involved in suing a governmental entity which is a specialized area of tort law that varies significantly from state to state.
It is likely that there is liability under a governmental waiver of the default rule of sovereign immunity in a case like this one, but it is also likely that special notices, court procedures and damages limitations apply to lawsuits brought to obtain this relief. For example, punitive damages are probably barred and other damages may be limited.
The primary statute relevant to a case like this one in Ohio is Ohio Revised Statutes §§ 2744.01-2744.11, a chapter which is entitled "Political Subdivision Tort Liability". ORS § 2744.02(B)(4) states (in the pertinent part) that:
[P]olitical subdivisions are liable for injury, death, or loss to
person or property that is caused by the negligence of their employees
and that occurs within or on the grounds of, and is due to physical
defects within or on the grounds of, buildings that are used in
connection with the performance of a governmental function . . . .
ORS § 2744.03(A) provides for a variety of defenses, most of which involve rogue employees, but one of which, (A)(5), provides that:
The political subdivision is immune from liability if the injury,
death, or loss to person or property resulted from the exercise of
judgment or discretion in determining whether to acquire, or how to
use, equipment, supplies, materials, personnel, facilities, and other
resources unless the judgment or discretion was exercised with
malicious purpose, in bad faith, or in a wanton or reckless manner.
There is an unusually strict form of two year statute of limitations set forth at ORS § 2744.04. The damages limitations are set forth at ORS § 2744.05. The liability and process for suits involving government employees is set forth at ORS § 2744.07. There are definitions at ORS § 2744.01 which mostly have the usual and ordinary meaning in the context of your fact pattern. Special procedures govern payment of judgments in cases like these if they are awarded. ORS § 2744.06.
Of course, like any other civil action, once a suit is commenced in a timely fashion by a lawyer in the proper court, there is a good chance that it will be settled before going to trial.
Substantively, the main question before the trier of fact at trial (states vary on the right to a jury trial in suits against them and I don't know what the rule is in Ohio) are as follows:
Did the school district (if the school district indeed owned the parking law) have a duty to use reasonable care to keep the parking lot free of ice?
Was the school district negligent (i.e. did it breach its duty to use reasonable care through its actions)?
Was this negligence the cause of the harm suffered?
What dollar amount is necessary to compensate the victim for the harm caused by the negligence, subject to the applicable damages limitations?
Other provisions govern how the public subdivisions are insured, and generally, in Ohio, the school district would be a separate entity with its own liability arrangements and insurance, and not under the town or municipality as @Cicero suggests in his answer. The way school districts are organized relative to other local governments where he lives is different than the way that it is organized in Ohio.