Just briefly looking at the error mentioning FK_ChildObject_AnotherObject... are you sure everything is wired up correctly in your EF data model concerning AnotherObject?
In your question you only list two tables, but this error is indicating that the ChildObject is participating in a 1 to * relationship with AnotherObject and there is not one present in the entity on save causing an error.
Try removing AnotherObject from the situation in order to test any assumed problems between ParentObject and ChildObject, or try changing the FK_ChildObject_AnotherObject relationship to 0..1 to * briefly for testing.