learning experience
Much of the feedback so far weighs in on the side of "required -- family is family". Someone cited honoring one's parents (the source of the request), and a couple people mentioned protecting a life (the sibling is apparently in real danger of injury without someone there).
I, on the other hand, am leaning toward "forbidden", though "permitted" is a possibility. Definitely not "required", though.
The issue is complex. While the sibling needs a caregiver, that's a service that can be hired -- so there's no apparent need for the poster to do it personally. Of course it's important to honor one's parents (this comes up a lot in text), but the talmud also teaches that if a parent asks you to transgress the Torah, you must decline (Bava Metzia 32a). This raises the question of whether attending another religion's worship service -- on its second-holiest day, to boot -- is avodah zara, forbidden worship. Is it enough if you don't intend to worship? What if you don't participate? What if you don't listen? That is a complex question with varied answers depending on circumstances, ranging from exactly what will take place to the strength of your own Jewish education and commitment, and you really need to ask your rabbi for a personal ruling.
I think the experience of facing this issue is valuable for the conversion candidate, actually. As a member of a minority religion (that sometimes faces hostility from others), sometimes you are going to have to make choices between your religion and your family/friends/society -- things like this, or resolving Shabbat issues with your employer, or various other matters. Finding out how you will handle those choices before it's "too late" -- before you convert and acquire new obligations -- seems useful to me.
I assume that most conversion candidates face some sort of religion-vs-world-at-large test during the process, but I don't actually know.

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I had a religion vs. world of large experience when I went to my niece's First Communion. I considered myself a guest (my family was aware of my status) and attended, but didn't consider it a form of idolatry. It was instructive for me because I discovered just how alien my Catholic roots had become to me.
I'm going to have a bit of a challenge this December because my sisters and their families are coming to spend the winter holidays with me. They've agreed to observe my dietary restrictions. We've yet to negotiate on the "celebration" of Christmas. I won't go to Mass with them, but on the otherhand if they want to open gifts for the girls and celebrate with a meal on Xmas Day, that's fine. There won't be a tree or any other Christmas decorations, that's something I'm going to be pretty firm on. Again, I will see myself as a "guest" but not a participant in the celebration
It's interesting negotiating those boundaries, but fortunately, my sisters are pretty understanding...
no subject
I visit my family in late December (generally a Sunday near the 25th) and we exchange gifts, because this is important to them. (I am selective in wrapping paper, by the way.) I don't participate in anything religious, and my parents have been pretty considerate about the music. (My father loves Christmas music...) They have also been very accommodating on the food front. They're trying to make me comfortable, so I'll honor their traditions to the extent I can. At this point it's really a visit, a meal, and an exchange of gifts.
Fortunately, we don't have any more small children in the immediate family; my sister's kids are in high school and college now. And they are so areligious that if they marry and have kids, the odds of something like a christening or a first communion even happening, let alone me being expected to be there, are minute. At this point, the only masses I expect to ever have to attend for family members are funerals.
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One way I'm dealing with it is my personal space "room" is emphatically Jewish. (I've lived in roomshares most of my adult life.) I have a mezzuzah on the door way, etc. I see open communal space that I share with my roommate, family and friends as being more neutral, hence flexible territory (i.e. I don't have a kosher kitchen, though I observe kashrut, etc.)
However, no Christmas decorations....it is a Jewish home, after all.
And no sappy Christmas music. I can deal with Bach...and Alvin and the Chipmunks...but that's about it...