Our Gemara on Amud Aleph implies that Gentiles are not specifically obligated in the commandment to honor thy father and mother. Likewise, Kiddushin (31a) indicates that even the notable Dama ben Nesina, whose stories of honoring his father were legendary examples and object lessons for the sages, was still technically not obligated in honoring his father.
A number of commentaries (see Yad Avraham and Rabbi Akiva Eiger on Shulkhan Arukh YD 241:9) raise objections to this formulation, as we see that Ham was punished for disrespecting his father Noach (Pirke DeRabbi Eliezer 23.) We also see that Rabbenu Bechaye (Devarim 21:13) holds there is indeed an obligation for a Gentile to honor his or her parents. The mourning time of one month for the captive woman, as described in the verse, is to allow her to fulfill her obligations to honor her parents who (likely) died the war.
Some offer an answer based on Rabbenu Nissim Gaon’s opinion (introduction to his commentary on Berachos), that Gentiles are obligated in “rational commandments” “מצות שכליות”. Rav Moshe Feinstein found this to be a forced answer (שו"ת אג"מ יו"ד ח"ב סי' ק"ל) and instead suggests that a Gentile’s obligation to honor parents stems from a principle of hakaras hatov (gratitude) which he says is universal and obligatory to all. (He cites midrashim which discuss lack of gratitude on the part of Adam and other biblical figures that predate the giving of the Torah.)
I will add my thoughts to explain Rav Nissim Gaon’s position. Just because one is not commanded in a particular law, it doesn’t mean that there aren’t consequences for behaving poorly. The whole idea of a commandment might be just a formalization of a particular expectation, and it does not mean that every other matter not stated is permitted by default, just because it’s not the law. This may be especially true in regard to Gentiles. It is one thing for a Jew, who has a covenant with God and 613 commandments, he can argue purely legalistically that matters outside the letter of the law are not obligatory. Granted, he might be a “נבל ברשות התורה“ “A disgusting person who behaves with technical permission of the Torah”, but his argument is still valid. While a Gentile, who has no covenant, might be more obligated to use his intellect to behave morally. This might be compared to the idea that Gentiles are not subject to specific thresholds (shiurim) in halakha, unlike a Jew. Therefore, a Gentile violates theft for less than a peruta, unlike a Jew, and Ever Min Hachai for less than a kezayis, unlike a Jew. (See Rambam, Laws of Kings 9:10 and also Annotations of Minchas Chinukh on Mishneh Torah, Laws of Kings.) Another idea, which could be combined in tandem with the above, is that since the requirement for making laws and courts is one of the Seven Noachide Laws, perhaps the core rationale is that self evident morality must be enacted and enforced. The law is secondary to the idea of the obvious moral principles.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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