The easiest way for you would be to allow users to add one dependent at a time. You present him with the same form as for filling in his own user details, except for username and password.
When the form is posted, you insert data into the person table. Then you get the id generated for that insert using mysqli->insert_id, assuming you're using mysql and the mysqli extension. This id is then used to insert into dependent with the user's id and the id retrieved above.
If they have more dependents to add, you just show them the same form again.
Optionally, you may want to let the user's search for other users to choose from as well. So rather than inserting a new person, they'd choose from an existing one, and when they submit from this form, you use the existing target's id and insert the user's id along with this in the dependent table. In case someone's kids are also registered users.
Furthermor, you might want to add a new field to the dependent table, request INT SIGNED DEFAULT 1, so that you can check this value when a user logs on and inform them that someone added them as dependents and let them verify if that is correct or not. If it is, you set request to 0, if they deny the request, you could set it to -1.