Is there a field on an item record that shows that it is a Parent Item? I would like to bring that into a saved search. I don't need to know the child item, I just need to know if it is a parent.

I believe it only reflects child - parent and not the other way. You would have to add a custom field and use workflow to populate it but at least you can identify the parent from the child so it should be straightforward.

You can also try using "Parent : Internal ID" -> "Is = None" in the criteria that should indirectly show the items that have no Parent (i.e., is a Parent Item). Another option is to use the "Parent" field or the "Parent..." join in the results to show which items have or don't have a Parent.