Combine varias filas en una MySQL Join

Puede utilizar GROUP_CONCAT para combinar todos los ID de acabado de una pieza determinada en una columna.

SELECT pj_items.part_num, 
       pj_items.qty, 
       GROUP_CONCAT(PartFinishes.FinishId) as FinishIds
FROM pj_items
JOIN PartFinishes 
  ON PartFinishes.PartNumber = pj_items.part_num
WHERE job_id = 1
GROUP BY PartFinishes.PartNumber

@comentario:Sin embargo, si desea mantener separadas las filas de pj_items, deberá unirse a PartFinishes ya agrupados por PartNumber:

SELECT pj_items.part_num, 
   pj_items.qty, 
   FinishesGrouped.FinishIds
FROM pj_items
JOIN 
  ( SELECT PartNumber, GROUP_CONCAT(FinishId) as FinishIds
    FROM PartFinishes
    GROUP BY PartNumber ) FinishesGrouped
ON
  pj_items.part_num = FinishesGrouped.PartNumber