What does the Catholic Church require for a parent's name to appear on the child's baptism certificate? Does the pastor of a parish, or his representative, have the right in canon law to prevent a parent's name from appearing on the baptismal certificate of their child to be baptized for any reason, such as lack of having received  all the sacraments?
Canon Law is explicit on this matter.

877 §1 The pastor of the place where the baptism is celebrated must carefully and without any delay record in the baptismal register the names of the baptized, with mention made of the minister, parents, sponsors, witnesses, if any, the place and date of the conferral of the baptism, and the date and place of birth.
877 §2 If it concerns a child born to an unmarried mother, the name of the mother must be inserted, if her maternity is established publicly or if she seeks it willingly in writing or before two witnesses. Moreover, the name of the father must be inscribed if a public document or his own declaration before the pastor and two witnesses proves his paternity; in other cases, the name of the baptized is inscribed with no mention of the name of the father or the parents.
[CIC]

The parents must be named; an unmarried mother or father must be named if the respective parent asserts that right in the manner required by the canon. If the infant's birth certificate names either parent (or both), that's a public record of parenthood and sufficient for the parent(s) who appear on it.