HPV is a major cause of cervical cancer, and the vaccine offers hope that fewer women will be killed by that disease. As such, it's pretty hard for anyone to come out against the vaccine head on.
Instead, the Dobson gang has decided to draw a line in the sand over mandatory vaccination. Of course, in the best tradition of a positive and open discourse, we really ought to say that they have decided to come out in favor of cervical cancer.
A successful vaccine doesn't just protect the recipient of the vaccine, it also protects the people around the target. When enough people in a society are vaccinated, diseases can't spread and epidemics are stopped before they get started. And getting enough people protected means requiring people to get vaccinated.
This falls within a broad category of situations that governement exists to address – tragedies of the commons. Each individual is better off not spending $360 to get a vaccine against a virus she may not get and which might not cause cancer if she did. So if everyone behaves selfishly, no one will get the vaccine.
The analogy to the historic example of the English commons system is obvious. For centuries, each town had a large area where everyone's sheep could graze. As long as everyone had about the same number of sheep and showed common care for the land, it worked fine. As herds got bigger, the common lands were overgrazed, and all the ranchers lost out because of the selfishness of a few.
The traditional solution to commons is to establish property rights. We each care for our own property because we are accountable for it. Systems like this can be established to remedy overfishing and other problems where property rights can be sensibly arranged.
Addressing disease transmission is not such a system. Short of imposing heavy fines for passing on STDs and regulating sexual activity (a tax on sex?), it's hard to know how to assign responsibility after the fact. So the government takes on the role of regulating use of the commons and fairly distributing the costs so that the benefits can flow to all. In this case, the best solution is to require all female children to be vaccinated. Any other solution will leave millions of women at immediate risk of HPV and therefore cervical cancer (and possibly colon cancer).