Yes it is, and yes they are. Morality is the basis of the law, and the government is the law in matters such as these. While the government should not indoctrinate people with morality, it is the fundamental tenet of a good society that its laws be consistent with sound moral principles. A law that mandates STD vaccines presumes the abundance of a lifestyle that spreads STDs, thereby conflicting with sound moral principles.
The notion that this vaccine is not forced is similarly a myth. It is forced unless you go through the legal process to obtain an exemption for your children. You are required to actively seek and obtain the exemption by the route prescribed in a government affadavit. Simply saying "no" will not exempt you, thus it is not true consent.
Why is the idea that each child's parents teach them morality and the government will make available vaccine for HPV to those who want it, such a bad idea in your mind?
You are arguing a strawman, and I suspect intentionally so. I have NEVER once suggested that simply making the vaccine available is a "bad idea" and defy you to show otherwise. I simply object to the government requiring school children to take it.
For the last time, the government isn't forcing anyone to take it. Any parent can opt out. And you accuse me of constructing a strawman?