Well that's a bit complicated. Traditionally, yes. Korea has been a Confucian society, which meant that the eldest son gets the parent's house, lives with them and takes care of them. Incidentally, being a wife of the eldest son is also very hard, because you are supposed to take care of your in-laws (boy do I have some horror stories).
With modernisation, this has changed.
1. Most women no longer wants to be servants for the in-laws (rightfully so) and wants to live separately
2. With less children, you won't necessarily have a son.
So now, parents live separately, until they are unable to do so anymore. When this happens, the children might take them into their house and care for them or admit them into hospice or retirement home.
You might ask, what happens if your children don't take care of you and you don't have money? Well you are kinda f-ed, as seen with the high elderly poverty rate.
> [Korea has] the highest elderly poverty rate among OECD countries
Oh, Korean national pension is also set to run out within 10 years or so, which will make the issue worse and/or make the future generation pay disproportionate amount of tax :)))
