Answer
Yes, the birth of a child in Uruguay can provide a basis for the parents’ residency, but the parents do not automatically receive passports. The child has their own status and documents, and the adults apply for their own status through the prescribed procedure.
For parents, local registration of the birth of the child and documents confirming kinship are important. The child's certificate may be used as a family basis where provided, but it does not nullify the identity checks, certificates, income and other requirements of the adult applicant.
If the parents already have another immigration case under way, do not discontinue it without a confirmed new route. A family relationship helps establish the basis, but each adult still receives an individual decision and a separate cédula.