If you are concerned that the Local Authority might turf you out of the house to raise the proceeds necessary to pay for care, using the 50% equity stake your Dad owns, it is unlikely to happen. They will not do this (generally) but ask for a revenue order to be signed such that they can reclaim their funding when the house is sold.
If you are concerned that your Dad's 50% stake in the property will not be left to you when he dies thus you are thinking about him transferring the whole asset to yourself, then it is unlikely to work. R1Geezer's answer applies.