PostgreSQL:busque permiso para el elemento, recorra hasta la raíz

Puede usar un RECURSIVE CTE por esto:

WITH RECURSIVE Perms(ID, Name, ParentID, CanRead, CanWrite, CanDelete) AS (
   SELECT i.ID, i.Name, l.LID AS ParentID, p.CanRead, p.CanWrite, p.CanDelete
   FROM Item AS i
   LEFT JOIN Permission AS p ON i.ID = p.ID
   LEFT JOIN Links AS l ON i.ID = l.ID
),  GET_PERMS(ID, ParentID, CanRead, CanWrite, CanDelete) AS (
    -- Anchor member: Try to get Read/Write/Delete values from Permission table
    SELECT ID, ParentID, CanRead, CanWrite, CanDelete
    FROM Perms
    WHERE ID = 3

  UNION ALL

    -- Recursive member: terminate if the previous level yielded a `NOT NULL` result
    SELECT p.ID, p.ParentID, p.CanRead, p.CanWrite, p.CanDelete
    FROM GET_PERMS AS gp 
    INNER JOIN Perms AS p ON gp.ParentID = p.ID    
    WHERE gp.CanRead IS NULL
)
SELECT CanRead, CanWrite, CanDelete 
FROM GET_PERMS
WHERE CanRead IS NOT NULL

El RECURSIVE CTE termina cuando un Permission el registro ha sido recuperado de la base de datos.

Demostración aquí