You need to be more specific. The mother must be here legally, or the father *if and only if* reliable DNA testing can establish that the child is, indeed, his. Otherwise, expect a rash of babies born of illegal alien mothers whose father is "Juan Pedro Gonzalez" (or "Muhammed Ahmed"), a legal green-card holder.
You are correct. Maternal citizenship or paternal citizenship DNA testing. (optional DNA testing for mom if in question)