Yes, parents are responsible for character education, because the children should be taught how to be good people. Children need to learn more than how to brush their teeth and some math facts. A parent has to teach a child how to go out in the world doing right instead of wrong.
Parents should definitely be held responsible for the actions and lives of their children. When a child is born, he or she is given no preconditioned thoughts about life. It only understands what he or she is taught. When parents teach a child good things, he will do good things. Likewise with bad things. Parents are responsible for a child's character as well as everything else.
Parents are responsible for character education. It is one of the most important things that a parent can pass down to their children. The school system also has a responsibility to teach character education. If a family attends religious services, they are also responsible to teach character education. The primary character education is left to parents.
I believe parents are responsible for character education. The term has been used to cover a lot of different things, but in basic terms it simply means learning to thrive and live in a social world without harming others. I believe character education is very important and it does fall partially on the parents, but school is also an important environment for this education as well.
While parents can add to an individuals thought processes, everyone is responsible for their own identity. Norms, values and individual character is gleaned from a persons environment, more than just one source. No one entity should be responsible for anyone else's outcome. All of us can add or detract in part, but it is never a good idea to do so in a totality situation. Eventually we would all be the same. Why would everyone in the world want to act like me?