Search recursively in MySQL5.7 table

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

https://dbfiddle.uk/?rdbms=mysql_5.7&fiddle=2f51944e354f92cfe7d428ce84f46184

Without a stored procedure, and with a maximum level of 3:

SET @search = 'param';

CREATE TEMPORARY TABLE Ptemp
   SELECT PARAM, P
   FROM (
      SELECT 
         PARAM,
         PARAM_VALUE,
         x.x,
         substring_index(substring_index(replace(PARAM_VALUE,')','('),'(',x.x),'(',-1) as P
      FROM parameters 
      CROSS JOIN (select 1 as x union
            select 2 as x union
            select 3 as x union
            select 4 as x ) x 
     ) x 
   WHERE P LIKE 'param%' 
;

CREATE TEMPORARY TABLE Ptemp1 as SELECT * FROM Ptemp;
CREATE TEMPORARY TABLE Ptemp2 as SELECT * FROM Ptemp;
CREATE TEMPORARY TABLE Ptemp3 as SELECT * FROM Ptemp;
CREATE TEMPORARY TABLE Ptemp4 as SELECT * FROM Ptemp;
CREATE TEMPORARY TABLE Ptemp5 as SELECT * FROM Ptemp;

select P from Ptemp WHERE PARAM=@search
union all
select P from Ptemp1 WHERE PARAM IN (select P 
                                     from Ptemp2 
                                     WHERE PARAM=@search)
union all
select P from Ptemp3 WHERE PARAM IN (select P 
                                     from Ptemp4 
                                     WHERE PARAM IN (select P 
                                                     from Ptemp5 
                                                     WHERE PARAM=@search))

see: DBFIDDLE

or DB-FIDDLE

The (multiple) copies of the temp table are needed because MySQL will give this error when re-using a temp table:

Query Error: Error: ER_CANT_REOPEN_TABLE: Can't reopen table: 'Ptemp1'

Conclusion: Writing a stored procedure is the way to go, at least in MySQL before version 8.0.