The procedure is that a birth must be formally declared at the municipality's office. The declaration can be made by anyone; in most cases the father of the newborn does it.
Furthermore two witnesses need to be present at the declaration; the person making the declaration can choose them freely. The birth certificate names the child and its parents, and it is signed by everyone present at the declaration.
In this particular case the mother (Aaltje de Vries) is not married, and hence no father is mentioned on the birth certificate.
Grandparents of the newborn are never mentioned on a birth certificate.