A: 

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.

I apologize for not included the third table in my brief overview. I have edited the original post to include the third table.Per your suggestion, I made the AnotherObject table 0..1. This changes the error to: A relationship is being added or deleted from an AssociationSet 'FK_ChildObject_ParentObject'. With cardinality constraints, a corresponding 'ChildObject' must also be added or deleted. This seems to imply that EF thinks I am trying to create another ChildObject... I even tried completely removing the AnotherObject relationship, but I still get the same error.
jslatts
Hmmm, well the good news is that AnotherObject is out of the picture now and the focus is where it belongs, which is the relationship between Parent and Child.Are you still producing this error by adding/deleting a Child? This error seems to indicate you're working on the Parent primarily. Like, you added a Parent, but the Parent/Children associate is * to 1 instead of * to 0..1. Adding a Parent in that scenario would demand the existence of a child, and potentially produce an error similar to yours.