Should parents be allowed to request that their child not be required to read parts of a history book in school that they find objectionable?

Yes, parents should have every right to request their children not read anything in school that they find shouldn't be allowed. They may have different religious beliefs, or just might not want their child exposed to certain language or topics. For instance, it would be wrong if a Jewish child was forced to read a Christian story in class without their parent's say in the matter. Same goes for a child who's parents don't believe that they should read about violence, racist, or sexual references. Parents have raised their children in a certain way, and may not want them to learn about things that go against what they have taught them.
Though children should have a right to chose in what they believe in, they will have the opportunity to completely make their own choices when they are an adult. Parents as legal guardians of their children, should have the right to chose what they take part in while they are in school.