The quote you replied to doesn't give the whole story though.
Yes, at age 11 a child probably could not give consent, but the HPV vaccination is given in Year 8 when children are 12 or 13.
By this age (and certainly by 14 or 15), many children will be competent to give their own consent and not require their parents consent.
However, most schools do ask parents for consent as well as otherwise they would have to assess the competence of each pupil individually.
But this does lead to pockets where uptake of the vaccine is low. It has been reported in studies since 2014 and here is a more recent study from 2021 from researchers at UCL as a pdf:
Which young women are not being vaccinated against HPV? Cross-sectional analysis of a UK national cohort study
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Don't forget that the courts have often ruled on children under the age of 16 being able to give consent for treatment.
This is usually referred to as "Gillick competency" (following a 1985 case of that name where a mother objected to her daughter who was under 16 being given contraception without her consent).
You may remember a court case from 2021, Bell v Tavistock. It went to the Court of Appeal and they decided that it was possible for children under the age of 16 to consent to receive puberty blockers if their doctor decided that they were competent to understand the situation.