Unless there are state schools sitting empty, with teachers but no students, to match the number of students in private schools, then those public schools are a subsidy on the state sector.No, there would not be. You have already stated that they have paid for the state school service in their taxes. So the resources are there for them already, they choose not to use them (except for the subsidies they receive and participation in the state examinations).
As you say, the parents have already paid for their children to be educated through their taxes. Because they send their kids to private schools they are not consuming those services. If someone pays my company for goods or services we do not have to provide that's a bonus.
The same goes for healthcare; people fund the states health service through their taxes but then buy that service from a private hospital. That is a saving to the state.
If there were no private schools tomorrow and the kids in those schools all want to state schools would the state have to provide more schools?
If there were no private hospitals or private beds in public hospitals and all those patients had to be looked after in the state funded facilities. That would add extra cost to the state system; private healthcare is a subsidy of the state system.