A convert is required to honor his parents. This does not apply when fulfilling their requests violates halacha. I suspect that many parents of converts, who are not knowledgable in halacha and probably don't care to be, would not really understand or accept a halachic response.
In the case of Christian or nominally-Christian parents, there can be tensions in late December. Whether Christianity is avodah zarah is controversial. One seemingly-non-religious aspect of their holiday is giving gifts, which is typically done at family gatherings (not privately). Leniencies appear to exist in the workplace, so one might say "if so, how much the moreso for parents?". While we are not supposed to give gentiles gifts before their holiday, it has been argued that we aren't necessarily concerned about prompting their worship by giving gifts (they don't tend to see the gifts as religious). On the other hand, and as noted there, there is also the consideration of imitating the ways of non-Jews.
Most of the converts I know are stumped at this time of year. Strict halacha seems to say they shouldn't participate in these family gift exchanges, but refusing to do so can cause upset (concerns of shalom bayit). In some families deferring to a different time of year (e.g. Purim, or emphasizing birthdays instead) is not acceptable to the parents.
So my question is: beyond consulting his rabbi (which he should of course do, though the rabbi probably doesn't know the family members), how should a convert weigh these various factors? How should he decide when to say "sorry it upsets you Mom, but no" versus going along with it? I am looking for practical answers moreso than halachic explanations.