SUBSTRING_INDEX(Дети, ';', i)

SELECT id, parent_name, SUBSTRING_INDEX(SUBSTRING_INDEX(child_names, ';', i), ';', -1) AS child_name
FROM mytable JOIN 
(
SELECT 1 i UNION ALL
SELECT 2 i UNION ALL
SELECT 3 i UNION ALL
SELECT 4 i UNION ALL
SELECT 5 i UNION ALL
SELECT 6 i UNION ALL
SELECT 7 i UNION ALL
SELECT 8 i UNION ALL
SELECT 9 i UNION ALL
SELECT 10 i
) t
ON SUBSTRING_INDEX(CONCAT(child_names, ';'), ';', -1) <> ''