If we can deport them or convict them of a crime that they commit, then they are subject to our jurisdiction. Ambassadors, consuls, diplomats, United Nations kakistocrats, and their families, employees, servants, and slaves fall outside our jurisdiction. Under international law, we cannot convict them of crimes that they do commit, such as illegal parking, enslavement, or even murder.
If the parents cannot legally be called citizens of the US nor are legally following our laws, they are nor here within our jurisdiction.
If the Soviet Union launched a nuclear weapon and discharged it over one of our cities, was it here by or under our jurisdiction?
Jurisdiction, as it is mentioned at Dictionary.com, can be substituted with the word "authority" or "control".
http://dictionary.reference.com/browse/jurisdiction