MySQL 5.7

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
select version();


version()


5.7.35


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

INSERT INTO parameters VALUES ('param', '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');
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))





P


param1

param2

param3

param4