Answer
A family relationship with a minor Uruguayan citizen may give parents a special family basis for applying for permanent residency. This does not mean that permanent residency is issued automatically at the maternity hospital: each parent applies and must confirm identity, the relationship, civil-status records, and other requirements.
The child’s birth certificate must be registered, and the parents’ foreign documents must be in an accepted form. Certificates, proof of address, and an in-person appearance may also be required. One parent’s status does not replace the other parent’s application package.
After the birth, first obtain the child’s Uruguayan birth record, then request the current procedure from Migración or the Ministerio de Relaciones Exteriores, as applicable. Do not buy a package promising “guaranteed permanent residency”: the decision is made by the state authority.