How to UPDATE a table for the level of parents in a column?
I would suggest to create stored function for that.

Table definition I've used for debugging:

DROP TABLE IF EXISTS `test`;

CREATE TABLE IF NOT EXISTS `test`(
id INT NOT NULL AUTO_INCREMENT PRIMARY KEY,
parent INT NULL,
parent_level INT DEFAULT 0
)ENGINE = InnoDB;

START TRANSACTION;

INSERT INTO `test`
    (`id`, `parent`, `parent_level`)
VALUES
    (1, NULL, 0),
    (2, 1, 0),
    (3, 1, 0),
    (4, 3, 0);

COMMIT;


Now, create a stored function:

DELIMITER $$

DROP FUNCTION IF EXISTS `getLevel`$$ -- drop if exists

CREATE FUNCTION `getLevel`(`anId` INT) -- argument: current row id
RETURNS INT                            -- returns int
NOT DETERMINISTIC
READS SQL DATA
BEGIN
DECLARE `aResult` INT DEFAULT 1;       -- result buffer, default value = 1
DECLARE `aParent` INT;                 -- buffer for parent ids

-- select first parent of this row:
SET `aParent` = (SELECT `parent` FROM `test` WHERE `id` = `anId`);

-- while parent id is set:
WHILE NOT `aParent` IS NULL DO

    -- increase result:
    SET `aResult` = `aResult` + 1;

    -- select next parent:
    SET `aParent` = (SELECT `parent` FROM `test` WHERE `id` = `aParent`);

END WHILE;

-- return result
RETURN `aResult`;
END$$

DELIMITER ;


Now, how to update:

UPDATE
    `test`
SET
    `parent_level` = `getLevel`(`id`)
WHERE           -- if safe mode is on
    `id` &gt; 0;   -- I have to use primary key column in update or it will fail

SELECT
    *
FROM
    `test`;


Results:

1   NULL    1
2   1       2
3   1       2
4   3       3