Assuming the hierarchy is just one level for parents and one level for children, we can try:
SELECT ud.name
FROM users_parent up
INNER JOIN users_data ud
ON up.children_user_id = ud.user_id
WHERE
up.parent_user_id = <some value>;
Edit: Based on this SO question I managed to piece together a hierarchical query which should work for your schema:
SELECT children_user_id, ud.name, parent_user_id
FROM
((SELECT * FROM users_parent
ORDER BY parent_user_id, children_user_id) parents_sorted,
(SELECT @pv := '1') initialisation)
LEFT JOIN users_data ud
ON parents_sorted.children_user_id = ud.user_id
WHERE
FIND_IN_SET(parent_user_id, @pv) > 0 AND
@pv := CONCAT(@pv, ',', children_user_id);
Demo
To use the abovr query, just copy it over to Workbench (or whatever tool you are using with MySQL) and then assign the parent_user_id
you want to the session variable @pv
. The output will contain all children descended from this parent, along with their names.