§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 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.
§3. If it concerns an adopted child, the names of those adopting are to be inscribed and, at least if it is done in the civil records of the region, also the names of the natural parents according to the norm of §§1 and 2, with due regard for the prescripts of the conference of bishops.
Source: Code of Canon Law (Vatican.va)