What is it called when a hospital refuses to put a tracheotomy tube into this child's trachea so the parents can take the child home?

Need more information: We can't answer this question without more detailed information about the case. But since there seems to be a disagreement between the doctors taking care of the child and the parents, this service is probably not a good place to get a second opinion. If you question the decision of the care providers, you may need to involve an ethic committee to review the case or request a second opinion from another specialist. Good luck, I hope the child eventually recovers!