Something kind of related to this that I've been thinking about. Consider these two propositions:
1) Vaccines mostly work for the diseases they are meant to prevent, and don't cause autism.
2) Vaccinations should be mandatory.
The first statement is objective and provable. The second statement is subjective and not provable. It's important to know the difference! I occasionally see someone trying to claim their opinions on objective issues are valid. But I think it's at least as common for someone to mistakenly believe they "proved" something about something that's actually subjective.
I would add a couple of propositions to link the two, myself. That is,
i) Vaccination is not just a means of protecting the vaccinated individual. It is a public health measure, and the state clearly has a legitimate interest in controlling and preventing the spread of dangerous diseases;
ii) In order effectively to control, if not fully eradicate, a particular disease, a sufficiently large percentage of the population needs to be vaccinated in order to build up "herd immunity";
iii) The number of people refusing, because they've been mislead by lies spread by either knaves or fools, to allow their children to be vaccinated has risen to an extent that it is putting at risk not only the health of their own children (which is bad enough) but also compromising herd immunity and thus causing a serious public health issue the threatens the health of many children and adults.
So, if necessary, vaccinations should be mandatory.
I would also argue (and it's uncontentious law in the UK) that the parent has the power to consent or not to a child's medical treatment only to the extent that that power is exercised reasonably and in the best interests of the child, rather than simply in accordance with the parent's particular religious or philosophical beliefs and prejudices.
If I want to go against sound medical advice when it comes to my own medical treatment then that's my absolute right so long as I'm able to understand the advice and the consequences of disregarding it. But if I'm acting on behalf of a minor child, then I don't have that discretion, because it is the child's health that I'm endangering, not mine.
Since it is not reasonable to refuse non-contentious medical advice from a competent medical practitioner, the parent cannot legitimately refuse to allow the child to be vaccinated, and the state has every right to intervene to protect both the child and public health in general.