Unlike most Mitzvot (Jewish religious commandments), Chanukah is not an individual Mitzva like Tefila or Tzedaka, but a family Mitzva. In some ways, similar (but not identical) to the Mitzva of lighting Shabbat candles, which is not done individually by each member of the family.
Illustrations: If one's son (or daughter) lives overseas, and he is financially dependent on his parents, he does not need to light his own candles. To this effect, a son or daughter is considered part of the immediate family while they are financially dependent on their parents (somekh al shulchan abiv). However, if they live in their own home and are financially independent (i.e., file their own Tax-return) they should light their own candles, even if they are still single.
If the husband is in a business trip, technically, he is included in the candle lighting done at home by his wife and children.
In both cases, if those who are away from home still want to light the candles away from home, they could do it, but without saying a Berakha.
If you are spending Shabbat in your parents' (or in-laws) home, you and your immediate family (spouse, children) are considered part of the extended family of your parents, since you also partake the same food, house, etc. So, when they light the Chanuka candles, your family is included in their Mitzva without further requirements. However, if you are going to arrive at your parent's house after Shabbat has begun, then you should light Chanuka candles at your own house.
In case you will leave the Chanukia lit at your house, you have to take extreme precautions to avoid any fire hazard.