In my country in East Asia, it is extremely impolite if I don't do it when I have this information.
However, the US and the EU are known to be more "professional", in my humble opinion. So, in the US, is asking the advisor about his father generally considered "impolite" or "unprofessional" from a professional point of view?
Unless you have some direct relationship with his father, this is not something we'd normally do in US, UK, or Australian culture. (I'm not sure if it's the same everywhere in the EU.)
A polite option might be to tell your advisor something like "I'm very sorry to hear about your father, please let me know if there is anything I can do."
If your advisor is from a different culture, that might change things.
The situation is the same in the EU as it is in the US. You should generally not go visit unless you are close to your supervisor's father personally.
If your supervisor has let you know about their father's situation, a reply similar to the one in Geoffrey Brent's answer is a good way to go. If you want to show polite interest, you can after some time casually ask about it, but quickly drop the topic if your supervisor replies vaguely or not at all.
I agree with other answers. Having said that, if the father does pass away, and if it is in the same town, you may want to go to the memorial service or funeral, as this is a way to show support for the living. Going with a group of students would also be fine.
To the excellent answers above, let me add that the situation is also the same in Russia, Georgia, Norway and Iceland. This previously posted answer would also be great there.
Please don't take this the wrong way, but I find it strange that you have to post this question on this site. Are there no local friends you could talk to? (If there is a different reason you ask this question, please see this comment as void.)
No, you shouldn't go to see your advisor's father, as per other answers.
The western equivalent is the sympathy card sent to your advisor (if you were very close to your advisor and had met the father in person before, you might send one to the father, too). A card with a short, handwritten note expressing your sympathy for the difficult time your advisor is going to. Stick to a serious card with a simple message. Avoid "get well" cards if there's a strong possibility that the father won't get well, but also avoid talking about loss or death since the father has not yet died. Avoid anything overtly religious, unless you know that your advisor is religious.
And if the father does die, then you should definitely send a card (a condolences card), plus there may be an office/department card, and possibly everyone will chip in to buy a flower arrangement for the funeral. The department may announce it, or you may have to ask the department secretary if any group thing is being planned.
I must offer a somewhat different answer. I will add that I agree with other answers that it is atypical, but we don’t know your relationship with your advisor or their family. This said, the following is going to assume you haven’t met their father. I disagree that you should absolutely avoid doing this and therefore must present that POV.
Depending on how you approach the matter, you may visit the loved one of someone that has been so influential on your life, such as an advisor.
If doing so, I would:
Now, I wouldn’t just show up out of the blue. I might also run the customs of your country by your advisor. First express your case and get their approval and ask for when a good time to visit may be. If you do visit, keep it short and sweet. People generally in an uncomfortable position don’t like like talking with people they don’t know.
One other note: although the common culture in developed countries does not do this, that does not mean that they didn’t in the past. The older generation sometimes have a better understanding of these customs and often welcome these gestures. To them, their life has passed by and they’re biding their time. They tell their stories to strangers on a park bench and they come from a time where boys swam together naked in gym class and guys openly changed in locker rooms - perhaps two weird examples for me to present, but they are striking difference that demonstrate how sensitivities have intensified in modern culture and just because “we” think something is indecent, does not mean it is to the subject.
Good luck with your decision. It is ultimately your decision and you must weigh your mind and your heart, which you’re doing a good job already simply by asking the question here.
