Who is responsible for the student's health and well-being,
Actually both parties. When kids are in school the district takes jurisdiction. Try going to a school and taking your kid out without permission. School employees are what are referred to as mandated reporters so IF they even suspect child abuse they are required by law to report it. And let's face it, there are probably teachers who care more about those kids than some of the parents in this community.
That being said, a very famous democrat who I loathe once said, "it takes a village" lending credibility to the claim that even a broken clock is right twice a day.