There is a thing called the "12 week disregard" where for the first 12 weeks the value of the house is not counted as an asset and as far as I'm aware going from LA funded to self funding triggers the disregard, however, you say he is only partly funded by the LA. Is it that his cash assets are between the £13k where the LA fund and the £23k above which you self fund? hence he is paying a %age towards the costs.
If so I believe you can ask for a 12 week disregard on the value of the house but LA's vary on their interpretation of the regulations or at any rate they try to.
Until, sadly, your mother passed away your dad didn't have the asset so I would ask for the 12 week disregard.
If you can rent a house out, cover all the costs of doing so and end up with more than the cost of a week or 10 days care then do it, I'd be surprised if you could in most of the UK though and a deferred payment whilst worth looking at is (in my view) just delaying the inevitable, sell the house and incurring cost in doing so but it does buy you time if you can do things to improve the value or saleability.
Sorry, I just noticed it's your first post so welcome to TP and my commiserations one the loss of your mum.
K