What does the Catholic Church require for a parent's name to appear on the child's baptism certificate?

The Code of Canon Law doesn't specifically mention a baptismal certificate anywhere. That said, a baptismal certificate is in intention a true copy of information present in the parish record. One would therefore expect the baptismal certificate to present all the information available in the official parish records.

Because Baptism is "the door which gives access to the other Sacraments" (Catechism of the Catholic Church, paragraph 1213),it is exceedingly important to the life of the Church that information about a person's baptism be properly recorded in a timely manner. (Otherwise, someone might be improperly admitted to, or more importantly denied, a sacrament.)

This information regarding the Sacrament is so important that Canon Law specifies what is to be recorded and when. Canon 877 specifies:

§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.

§2 If it [i.e.the baptism] 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.

In other words, the name of the parent must be recorded, if either

• the parents are married, OR
• the parents are not married, but either is identified in the public record (e.g. on a birth certificate, in a hospital medical record, or even in a published birth announcement), OR
• the man or woman formally states that they wish to appear as father or mother in the parish record. (The specified formal statement differs for a man and a woman.)

It should be pretty obvious that it's rather difficult for a parent's name not to be included in the record, and thus on the baptismal certificate.

A related question, which was posed in an earlier version of this question, is whether the parish can delay putting the parent's name into the record, for example until a catechetical program is completed. This is clear, however, from section 1,which states that the pastor must enter information into the records "without delay". This generally means within 24 hours, unless the pastor is for some reason unable to access the parish records. In particular, it is a violation of section 1 for the pastor to delay completion of the record (and thus of the baptismal certificate) until a parent has fulfilled some requirement of the parish (receiving baptism or confirmation, for example).

My answer then is, Except in very rare cases, both parents' names must appear in the parish record, and in the baptismal certificate which reflects the record; and this entry must be done as soon as possible after the baptism, without requiring the parents to fulfill any other requirements first. Failure in either of these counts constitutes a breach of canon law and should be reported to the pastor or to the bishop..

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.