Ah - then my original suggestion will give you exactly that. However, I might have confused you - instead of
SELECT <desired columns>
FROM my_table
WHERE child_column <> parent_column;
I should have said
SELECT <desired columns>
FROM my_table
WHERE child_column <> parent_column_value;
You would supply the parent's value when you run the query.
However, you confuse me with your response:
"Your suggestion would likely get me all of the children related to other parents, not those related to the children themselves."
I didn't see where there was a requirement to relate children to children.
But if what you are looking for is "a list of children (which may or may not be related to other parents) but are not related to a given specific parent", then why is a query that will "get me all of the children related to other parents" not what you are looking for?
The only other thing I can see is if you have some children that have a null value in the column(s) that form the foreign key relationship. If that is the case, then you could try the following:
SELECT <desired columns>
FROM my_table
WHERE child_column <> parent_column_value
OR child_column IS NULL;