Серийный софт не умеет писать и читать данные

Попробуйте это:

DELIMITER $$
CREATE TRIGGER `trg_part_upd` AFTER UPDATE ON `tbl_sub_model_eng_trans_part`
 FOR EACH ROW BEGIN 
    IF NEW.engine_sk = 0 AND NEW.trans_sk = 0
    THEN
        UPDATE tbl_sub_model tsm
        INNER JOIN tbl_fct_sub_eng_trans tfset ON tsm.sub_model_sk = tfset.sub_model_sk
        INNER JOIN tbl_sub_model_eng_trans_part tsmetp ON tfset.trans_sk = tsmetp.trans_sk
        SET tsm.last_modified_date = NOW()
        WHERE tsmetp.sub_model_sk=NEW.sub_model_sk;
    END IF;
  END;
$$

DELIMITER ;