Who should play the larger role in a child's education: the child's parents or the school?

Parents play the large role in a child's education. The child can be in the school for just 4-6 hours. Parents are with their child for above 18 years. 

Interactions between teachers and their students will vary depending on the child's developmental and behavior attributes. There are other factors to consider: Time, income, availability, and a healthy educational environment. If a teacher for example is abusive in their methods of teaching, then having them play a larger role towards a child can be detrimental to that child's progress in school. 

Then the factor of parents and their involvement: Parents have a fundamental role in being present in their child's care and development. It is not up to teachers to ensure that a child is encouraged at home to be intellectually curious. A teacher's influence can only be a positive one if the parents themselves are very much involved as well, and open to new ideas and new methods of teaching.

So the answer is that both teachers and parents must be engaging and present in the child's education. Taking an interest and being a positive enforcer, rather than a pressure conductor or someone who discourages a child's intellectual curiosities.