Parental supervision or guardianship does not trump the rights of children. Kids have a right to be educated, and they have the right to grow up in an environment that is free of abuse or neglect. Once a parent abuses these rights, they lose all rights to be a parent.
Children have the same basics rights that we all do as humans - life, liberty and the pursuit of happiness. Some other rights that are granted by governments, like voting, driving, area age specific rights and so children would not have those, but any rights that are considered basic to being human are shared by children.
I believe that rights do apply to kids. They are human beings and should have a right to decide matters for themselves. Without rights,kids could be taken advantage of very easily. I also think it depends on the children's ages to determine what decisions they can make such as what age to leave home , etc.
You could say that human rights such as freedom of expression, freedom of religion, freedom of speech and the right to privacy only apply to adults. What really is an adult? Age is just a number. I think that an "adult" is someone with an education. Someone who knows when to stop!