Can non-Catholic parents ask the priest to baptize the child (infant) in the Catholic church and promise that they will give the child a Catholic upbringing? I mean, it could be a learning experience for the child. Even if the child doesn't believe, such insider exposure may give the child some underlying presuppositions that Western atheists make (i.e. existence of God, the causal relationship between the supernatural and morality, etc.), which may not be accessible to atheists who are not raised in a Christian household. Also, even if Christian morality isn't perfect, the church at least provides a social environment as an attempt to not only teach morality in the Western world, but also promote community service, and Catholic churches are pretty common in the United States, so maybe the non-Catholic/non-Christian parents may willingly take advantage of that. Plus, I use the Catholic church in this question, because I know that the Catholic church seems to consider all people who are baptized are Christian, but not all Christians (in this sense, Catholics) are saved and that salvation is an ongoing process. I think that would be doable for a non-Catholic person who intends to raise his or her child Catholic. Even if the child may not become an observant Catholic in adult life, he or she will forever be Catholic, and the Catholic upbringing may be beneficial.
The Roman Ritual prescribes
24. An infant of infidel parents may be baptized lawfully even though the parents are opposed, provided that its life is in such danger that one can reasonably foresee it may die before attaining the use of reason. Outside the case of danger of death, it may lawfully be baptized, provided its Catholic rearing is guaranteed, as in the following two cases: (a) if parents or guardians or at least one of them consent; (b) if parents, i.e., father, mother, grandfather, grandmother, or guardians do not exist, or if they have lost their right over the child or are unable to exercise it.
25. Generally, the norms stated in the preceding rubric are to be applied to baptism of infants whose parents belong to a heretical or schismatic sect, or of Catholic parents who have lapsed into apostasy, heresy, or schism.
Note that "Catholic rearing" must be "guaranteed", and it is the priest who must assess that. He may feel that "infidel parents" may not be able to guarantee it sufficiently, even with a fully-qualified sponsor.
The Ritual recognises that the sponsor (or "godparent") will be an important part of the child's religious upbringing:
38. It is the duty of sponsors by reason of their position ever to regard their godchild as a personal charge, and in all that pertains to his Christian upbringing to watch over him faithfully, so that in his whole life he may prove himself true to the promises which they once solemnly spoke for him.
... and it lays down rules which need to be met in order that the sponsor is deemed capable of discharging that responsibility:
33. There should be only one sponsor (who may be of different sex from the one baptized); or at most two may be employed, a man and a woman.
34. To validly act as sponsor it is required:
(a) that the person is baptized, has attained the use of reason, and has the intention of acting in this capacity;
(b) that he does not belong to a heretical or schismatic sect, is not excommunicated whether by condemnatory or declaratory sentence, nor legally infamous, debarred from legal acts, nor a deposed or degraded cleric;
(c) and that the person is not the father, mother, or spouse of the one baptized;
(d) that he is chosen by the one baptized, or by the parents, guardians, or, if these are wanting, by the minister;
(e) that during the act of baptizing the sponsor (or his proxy) physically hold or touch the one baptized, or immediately lift him out of the water, or take him into his arms from the font or from the hands of the minister.
35. To lawfully act as sponsor it is required:
(a) that he has reached the age of fourteen, unless the minister sees fit to admit a younger person for some valid reason;
(b) that he is not excommunicated for a notorious crime, nor excluded from legal acts, nor legally infamous (even though no sentence has been issued to that effect), nor interdicted, nor a public criminal, nor infamous in fact;
(c) that he knows the rudiments of the faith;
(d) that he is neither a novice nor a professed religious, unless necessity urges it and the sponsor has the express permission from at least the local superior;
(e) that he is not in sacred orders, unless he has the express permission of the Ordinary.
36. When in doubt as to whether a person may validly or lawfully be permitted to act as sponsor, the pastor should consult the Ordinary if time allows.
(In every place in the quotes, lawfully is referring to Canon law. If the parents are opposed and a baptism takes place, there may be a civil case to answer.)
While some of the advice given above is similar to current canon law on these matters, not all of it is the same. The Roman Ritual from 1964 DOES NOT reflect current Church law, but apparently summarizes the 1917 Code of Canon Law. It is a head scratcher why it would be used as a source to respond to this questioner. What is relevant is the 1984 Code of Canon Law, which superceded the 1917 one (and doesn't label non-Catholics "infidels"). Moral of the story: if you try to get information off the internet it is very hard to be sure of its quality unless it's clearly from an official source.
Under any ordinary circumstances, non Catholic parents will not be permitted to have their child baptized in the Catholic Church, because it is not reasonable to expect that non-Catholics will raise their child to be a believing and practicing Catholic. Having Catholic godparents does not substitute for having at least one Catholic parent. Being Baptized in the Catholic Church means that the one Baptized is subject to Catholic Canon Law and all obligations of practicing the Faith. If the priest knows the child will be raised by parents who refuse to become Catholic themselves and therefore will not give an example and formation of that kind to the child then there is no basis for thinking the child will grow up to be able to fulfill that. So, the parents need to set out prayerfully on a journey of truth seeking, learn about the Catholic faith, and approach their local parish about becoming Catholic themselves, or else get the child Baptized in a protestant church. Sometimes the reason why non-Catholics want Catholic baptism for their child isn't even religious, but the parents think this will help their child get into a good Catholic school--this is an unacceptable reason, and frequently Catholic schools accept non-Catholic students anyway. Maybe when the child is older he or she will want to ask in sincerity to become Catholic.