MySQL 5.7

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CREATE TABLE parameters (PARAM VARCHAR(255), PARAM_VALUE VARCHAR(255));

INSERT INTO parameters VALUES ('param0', '1 + $(param1)');
INSERT INTO parameters VALUES ('param1', '1 + $(param2) + $(param3)');
INSERT INTO parameters VALUES ('param2', '1 + $(param4)');
INSERT INTO parameters VALUES ('param4', '1 + 3');
INSERT INTO parameters VALUES ('param3', '1 + 2');
CREATE PROCEDURE get_params_list (param VARCHAR(255))
BEGIN
CREATE TABLE params_list (param VARCHAR(255) PRIMARY KEY,
                          param_value VARCHAR(255)) ENGINE = Memory;
INSERT INTO params_list 
SELECT *
FROM parameters
WHERE parameters.param = param;
REPEAT
    INSERT IGNORE INTO params_list
    SELECT parameters.*
    FROM params_list
    JOIN parameters ON LOCATE(CONCAT('$(', parameters.param, ')'),
                              params_list.param_value);
UNTIL NOT ROW_COUNT() END REPEAT;
SELECT params_list.param 
FROM params_list
WHERE params_list.param <> param;
DROP TABLE params_list;
END
CALL get_params_list('param3');
CALL get_params_list('param1');


param


param2

param3

param4


CALL get_params_list('param0');


param


param1

param2

param3

param4