MySQL 8.0

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CREATE TABLE test (
  col JSON
);

INSERT INTO test VALUES
  ('{"parent":[ { "child":"1,3", "type":"" },{ "child":"36,41", "type":"1" },{ "child":"52", "type":"1" } ]}'),
  ('{"parent":[ { "child":"2,3", "type":"" },{ "child":"36,41", "type":"1" },{ "child":"52", "type":"1" } ]}');

Records: 2  Duplicates: 0  Warnings: 0

SELECT * 
FROM test
WHERE FIND_IN_SET(1, REGEXP_REPLACE(JSON_EXTRACT(col, '$.parent[*].child'), '[\\[\\]\\" ]', '')) > 0

col

7b22706172656e74223a205b7b2274797065223a2022222c20226368696c64223a2022312c33227d2c207b2274797065223a202231222c20226368696c64223a202233362c3431227d2c207b2274797065223a202231222c20226368696c64223a20223532227d5d7d

SELECT *
FROM test
WHERE JSON_SEARCH(col, 'one', '%,1,%', NULL, '$.parent[*].child') IS NOT NULL OR
  JSON_SEARCH(col, 'one', '1,%', NULL, '$.parent[*].child') IS NOT NULL OR
  JSON_SEARCH(col, 'one', '%,1', NULL, '$.parent[*].child') IS NOT NULL

col

7b22706172656e74223a205b7b2274797065223a2022222c20226368696c64223a2022312c33227d2c207b2274797065223a202231222c20226368696c64223a202233362c3431227d2c207b2274797065223a202231222c20226368696c64223a20223532227d5d7d