The relationship between rights and responsibility is pretty clear. As people gain rights, they gain corresponding responsibilities. Obviously someone is responsible for any action that is a criminal behavior. Lets look at trespassing, for example. It is a fairly simple crime and corresponds easily to a right.
Ask yourself, can a child choose where they go, legally or is that privilege controlled by their parents? Can a twelve year old child tell their parents they are going to the store and do so without permission? If a parent stops them will the police interfere to protect the child’s right to liberty? If the parents punish the child for going without permission by locking them in a closet for 4 hours, will the police interfere to protect the child?
At this point you probably realize that a parent or legal guardian hold a child’s right to go where they want as a legal trust for the child doling it out as they see fit. They can, for example, tell a child to climb over a fence and go pick some apples. They can punish them for not doing so. They can control through observation, punishment, and physical force where a child goes and be perfectly within their legal rights. How then, can you hold a child legally responsible for going anyplace? They have no legal right to control where they go, so they should hold no legal responsibility. The parent does hold legal right to control where a child goes and thus should hold legal responsibility.
There are few instances where this does not apply. Legally, a child does have a few rights. They legally have a right to live. A parent cannot execute their child. The same goes for battery and sexual assault. That is pretty much it though.
In a perfect world this would not be an issue. Parents would look after children and wisely administer their rights. This is not a perfect world. If a child steals my wallet and spends all my money, their parent can and should be held responsible for compensating me and making good on the crime. Their child has no right to own money or possessions and should not be held legally responsible for theft of either until they do have that right.