Question: Must I have my children vaccinated?

"I believe vaccinating children is dangerous, and is also unnecessary for me because all of the other parents have already had their children vaccinated."

The fallacy here is not in the beliefs and misinformation you cite ( or do not cite) to support your claim, but in the self-contradicting terms with which you state them.

If you truly believe that vaccinations are dangerous. you immediately contradict it by also "believing" your own child to be safer because other parents have already had their children vaccinated." This establishes a Contradictio in Adjecto or self-contradiction, self-refuting idea based on the whimsy of your own belief systems. So therein is the error in reasoning.

If you truly believe vaccinations are dangerous then it might also suggest your children could be less safe in socializing with vaccinated children. It all depends on what danger you're talking about. Of course, we can assume and generalize but that puts us all in a paralogical predicament that is not conducive to proper ratiocination.