Buying a house from parents

Yes, if you buy the home for less than market value, the nursing home or state health age funding your father's care (assuming he does not pay for it all out of pocket) would have the right to seek the difference in value and could do so by putting a lien on the home or undoing the transfer to you and forcing its sale. To force people to pay for their own care (instead of making the taxpayers pay), the law allows any transaction, transfer, sale, etc. which is done for less than fair market value to be voided, so that people cannot "hide" their assets and avoid using them to pay for their own care. This is not the time to purchase the home for less than FMV. Mind you, you can buy get it and get a good deal, but is has to fall within the following parameters:
1) Homes don't have a single FMV: they could reasonable\y sell for a range of prices (e.g. a home appraising for $110k could reasonably sell for, say, $99k to $120k+, depending on the market, how motivated the buyers are, whether the sellers are able to wait for the perfect offer or not, etc.). You could reasonably buy the home for near the bottom of the reasonable price range for it.
2) Since there is no realtor fee if a direct sale to you, you could knock off a few percent for the savings in commisions.
3) If the home can be shown (such as by a professional home inspector) to need a few thousand dollars of repairs, you could reasonably buy it for a slightly reduced price to reflect that, since otherwise, your father would have to put money into the home to repair it.
So you can get the home for a good price, but it still needs to be a fair price given the circumstances, and within the bounds of what a 3rd party, buying it under similar conditions (e.g. without a realtor; taking it without requiring repairs) would pay.