Interesting.
I am not a solicitor. This is not legal advice - just a summary of the working principles.
An action like this would be based primarily on negligence. To succeed in a negligence case the plaintiff has to establish 4 proofs ;
1. The defendant owed the plaintiff a duty of care. This is a matter of law as distinct from the evidence alone.
2. The defendant breached the duty of care.
3. The plaintiff sustained injuries and or losses.
4. The defendants negligence was the proximate cause of the plaintiff's injuries and losses.
Point 1 is a matter of law to be decided by a judge. The relevant concepts that would be taken in to consideration would be those of reasonable foreseeability and what lawyers know as the neighbour principle.
The concept of the neighbour principle is that those to whom a duty of care might be expected to be owed would be those so closely and directly affected by my actions that I ought reasonably to have them in consideration as being so effected when I am directing my mind to the acts or omissions that are being called in to question. See Donoghue -v- Stevenson (1932).
If a judge decides that no duty of care is owed the case fails at that first hurdle.
Even if a judge decides that there is a duty of care owed proofs 2, 3 and 4 above still have to be established and those are mainly issues of fact based upon the evidence.
Offhand the threat sounds a little bit remote but it could be arguable.
Basically, a plaintiff would have to establish that the parent knew or ought reasonably to have known that failure to inoculate / vaccinate would cause another child to be infected. Ultimately, if a duty of care was held to exist much of the case would turn on the facts.
For example, knowingly sending a child to school whilst in a highly infectious state clinically could ground the proposition that this was unreasonable behaviour.
I would notify the personal / public liability insurers of the parents in question just in case that even a speculative case is launched.
