Get Immediate and Ultimate Parents with levels

WITH r (id, parent, ultimate_parent, name, ultimate_parent_name, lvl) as
   (SELECT id, parent_id AS parent, parent_id AS ultimate_parent, name, name as ultimate_parent_name, 0 lvl
    FROM mytable
       WHERE parent_id = id -- identifies a root
UNION ALL
    SELECT m.id, r.id, ultimate_parent, m.name, r.ultimate_parent_name, r.lvl + 1
    FROM r join mytable m on m.parent_id = r.id  -- joins child with parent
    WHERE m.parent_id <> m.id -- to avoid cycles
   )
SELECT * FROM r ;

The first part of the subquery fetches the roots and the second part connects 

the children.  

Parent is the immediate parent and

ultimate_parent, the ultimate parent.