This is somewhat of a difficult question to answer, because while the child is the parents responsibility, they may very well not be trained professionals like a Doctor. The Doctor will most likely know what is medically best for the child, but ultimately, it is completely up the parents to decide.
Parents make almost all decisions when it comes to their child's life. They influence what they eat, what they wear, where they sleep, what they watch, and what they learn. This should also be true of medical care. Unless a parent is in danger of harming the health of their child, parents should be able to make informed medical decisions for their children.