INNER JOIN same table

I think the problem is in your JOIN condition.

SELECT user.user_fname,
       user.user_lname,
       parent.user_fname,
       parent.user_lname
FROM users AS user
JOIN users AS parent 
  ON parent.user_id = user.user_parent_id
WHERE user.user_id = $_GET[id]

Edit:
You should probably use LEFT JOIN if there are users with no parents.