As well as halachic answers, there is this point to consider:
It is a very important command, to honour ones parents, in most circumstances. To deny a person's fact of being a parent, at a pinnacle of parenthood (when ones child is themselves becoming married) would seem to be a height of disrespecting them as a parent.
Similarly one should also not place a child (or child's spouse) in the position of being expected/required/socially pressured to take or permit that act of disrespect to their parent. This would be not only a direct breach, but also a further breach of another commandment, not to place a "stumbling block" before "the blind".
Update: Also one is required by both positive and negative commands, and rabbinical law, to take all efforts to avoid singling out and humiliating a person in public, whether by deliberate act or omission - an act which is compared to killing them. Chesed (lovingkindness) is one of the highest virtues cited in Jewish law, which is full of positive examples where the strict letter of law is overridden by the demands of chesed. Singling out the parent, especially on a day like their child's fulfilment of such an important commandment as marriage, would seem to be a prime example of breaching that requirement also.
(It shouldn't need saying, but all of these apply to any people (not just family and community). That's not just because of natural justice and ethics or because it would be plainly abhorrent not to. It's also shown explicitly by rabbinic examples and anecdotes, which show the application of these same laws with regard to both jews and non-jews.)
I've been asked in the comment to add support for the obvious points, which is fair. I didn't expect some of them to be points for which support might be needed, being secondary points that are non-contentious, fairly well known, and with luck, extremely obvious. Here are a few - possibly not the best and surely not all that exists:
- Honour due to ones' biological parent if non-jewish: (1) See the Talmudic story of Dama ben Nesinas (Kiddushin 30a), the non-jew who honoured his parents and was highly praised and, the next year, divinely rewarded, apparently for doing so. (website discussion) Striking that the principle to honor parents is illustrated by a story of a non-jew honouring his non-jewish parent. If it is praiseworthy for a non-jew with non-jewish parents to honour them, so much more is it for a jew with a non-jewish parent, lest it be thought that adopting any less respectful style is inherent in conversion, or the like.
A related question is whether a convert should still honour their parents after conversion, discussed on Chabad. Same reason as above also applies, though this is hardly the "reason" to do it.
- Honour due to parents-in-law: Widely cited verse Samuel 24:12. Even though "father" can be a term of respect or for an elder, it's generally accepted that this teaches us that a father in law is also to be treated to a great extent as a father. (And a mother in law therefore to a great extent as a mother). At least one website states this is also codified in Shulchan Aruch. The discussion does not seem to distinguish between parents-in-law that are jewish or not, or err or no; as a general principle, in deciding upon the existence of a requirement to honour them.
- Comment on the two together: Apart from logically combining these to discover our duty of honour to a non-jewish parent-in-law, one can also discover further separate reasons. For example, let us suppose that we accepted that the biological child is directly required to honour the parent, for the reason above, but then consider whether their partner should reject the parent being treated this way, because the two above items can't (for any reason) be combined this way. But that would require the partner to demand their betrothed partner breaches their duty of respect to their own parent. That would probably be inconceivable, due to the same breaches which it would cause them to commit, including the nature of their conduct towards and harm done to their own spouse (and his/her relationship with his/her parents) in doing so, and Shalom Bayith both in their home and at the wedding. Also we would note that the biological child has an explicit duty to honour, while the other has no comparable explicit duty to forbid that honouring.
As an aside worth noting, the duty to "honour" ones parents, covers an intent and the steps needed, to make them feel honoured. That would be breached by excluding them. It is also a general rule that if there is doubt, one should err on the side of caution, lovingkindness, and more eagerness to fulfil the important commandment to honour ones parents (jewish or not, in-laws or biological), to seek peace and to not cause Hilul Hashem (as added in comments), and to not place a stumbling block.
But mainly, one should seek to "be holy" as many commandments flatly state, and one should consider what a truly godly person filled with divine chesed and compassion and love, would do.
Although a personal feeling and not a statement of law, I do feel that is all the reason a person and their family and community should really require, for the decision to welcome their in-laws warmly and with deep respect and love, to their wedding, and grant them all honours that are not explicitly forbidden on the highest authority of an explicit negative commandment to grant them, as befitting the most blessed biological (or indeed any other) parents of one of the marrying couple on that day. The rest (to paraphrase Hillel) seems to me, to just be explanation.