>Should this do it:
>
>

CREATE TRIGGER PersonsDeleteDependents 
>   ON  Persons
>   FOR DELETE
>AS 
>BEGIN
>	-- SET NOCOUNT ON added to prevent extra result sets from
>	-- interfering with SELECT statements.
>	SET NOCOUNT ON;
>
>    -- Insert statements for trigger here
>    delete Persons
>    from Persons
>    inner join deleted on deleted.personid = persons.parentid
>
>END

>
>?

The trigger looks OK, but did you have recursive triggers enabled?
How many levels of parenting you may have?
What if you delete a child that has siblings?