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As far as I'm aware, South Korea does not have a documented limit on how many days per year you can stay.
Per anecdotal evidence, doing repeated "visa runs" to renew a 90-day tourist visa appears to be generally acceptable, but expect to get asked what you're doing in Korea and how you're planning to support yourself sooner or later. If you do get denied entry, that will considerably complicate any future visa applications, so you're definitely best off looking into a legitimate long-term visa if you need to look after your family.
Upvote:1
I spent a few years in Seoul in the 1990s on visa runs, before I managed to get a visa. Back then it was acknowledged, and tolerated. Although the colour of your passport and skin influenced that.
It has become harder, for various reasons, among which (in the eyes of Immigration) too many foreigners abusing the system, and the diversification of possible visas (which is viewed by authorities as an improvement, ie, there are now enough visa categories to fit all cases).
One thing that has worked well for me was to travel often, never using the full 3 months every time. I would go to Japan, mostly, on a regular basis. However, that too came with its own sets of problems, as airlines in Japan are very careful about letting people board a plane to a country where they're not resident.
Bottom line, you should be able to stay a while, with visa runs, but if you ever get questioned by Immigration, be prepared to be refused, or given a shorter stay, the next time. Korea Immigration keeps a record of all entries. I had to get my record once, for visa purposes, and it had everything...
If that happens, and you need to stay longer, enlist for a Korean class.