I don't understand why the parents didn't apply immediately while still in the US? AFAIK this sounds correct - if the child has diplomatic status thru the parents they are not eligeable for citizenship. I would check the website of the US passport agency/Dept. of State. Certain semi-diplomats (e.g. people working for international organizations such as the World Bank) don't have full diplomatic status and their kids do seem to be eligeable - so this angle may be worth checking out.

.....The 14th Amendment of the U.S. Constitution guarantees citizenship at birth to almost all individuals born in the United States or in U.S. jurisdictions, according to the principle of jus soli. Certain individuals born in the United States, such as children of foreign heads of state or children of foreign diplomats, do not obtain U.S. citizenship under jus soli.