Schools have many resources available to use to broaden the cultural awareness of their students. They have access to information, guest speakers, field trips (virtual and physical), as well as the cultural differences within the student and faculty in the school system. Parents can help some with cultural awareness, but they do not have the resources that the school has.
I think the schools should be responsible for the cultural awareness of children. Cultural awareness could be incorporated into their existing curriculum without any major modifications. Since schools already teach geography and history, it would be appropriate to include cultural studies for each area covered. It might also be good if they could start a separate class simply for cultural awareness in today's society, especially sine the US culture is so diverse.
Yes, schools should be responsible for the cultural awareness of children, because we can't leave it up to the parents. The children need to learn that the world is bigger than just their school. They need to learn that people in the rest of the world have different circumstances than them.
Not just for "cultural awareness", but for everything. As the parent, it is their duty and responsibility to have first say in everything concerning the upbringing of that child. At best, schools can be considered 'facilitators' of education, but even a child's education should include active participation from the parents. A child trusts their parents more than anyone in the world, and (most) mothers especially are more close to their children than anyone will ever be.
I believe that parents should be the ones responsible for their children's cultural awareness. I do not think that schools should be responsible for such a thing. Schools should just be responsible for teaching kids about educational programs and fields. Everything else that deal with society is up to the parents.