The jurisdiction language basically excludes parents with diplomatic immunity, under current rules, not other people flying above the US regardless of the aircr

The jurisdiction language basically excludes parents with diplomatic immunity, under current rules, not other people flying above the US regardless of the aircraft's country of registration or any other status attributes of the parents. In practice, what I said above is well-settled legal and administrative practice in the US, whether or not the constitutional words could have been interpreted or differently by Congress and/or the courts. US authorities receiving proof of birth above the US and lacking reason to suspect parental diplomatic immunity certainly will and actually do grant proof of US citizenship, no further argument needed once the location of birth is documented.