Up until two years ago, I was the main carer for my dad. Then my sister moved back into town. He has a clear preference that she provide all of his care and increasingly makes decisions with him. To put it simply, they get on better. When support is offered by anyone else, he just says no or I’m fine. The issue is that although this was ok at first, my sister can’t now meet all of his needs all of the time. She doesn’t want to and for the record I don’t expect her to. For her part she doesn’t want to say no but also she doesn’t see herself in the carer role. It’s difficult to bring up future planning or external support with either of them inc. a gardener, cleaner or meals service. When i do pick up some of the caring, I do a good job but my father makes it clear that he’d rather support from her and then my sister tends to criticise. I come out feeling rejected and concerned. I dare say that they see me and others as intruders in a well balanced system.
Dad and sister need to sort this out between them, honestly, that he needs more and she doesn’t want to do it all. IF you are happy to help, that needs to be accepted graciously by dad, no drip feed criticism, that’s cruel! It’s really time someone else, a cleaner/ cook took over some of the donkey work. You and your sister are his daughters, not slaves!
If she doesn’t want to care, that’s fine, as you said. But she has to tell him. It’s not about saying no, it’s about living her own life. And that’s how she (and he) needs to see it.
On the favouritism issue: parents rarely admit to it but there will always be favourites. It can be difficult - even damaging - when you know that to be true, and especially when there is no effort to hide it or compensate for it. Question (no need to answer, this is for you to think about): does your sister criticise you in front of your Dad to appease him, or behind closed doors? Clearly she admitted to you that she doesn’t want to tell him about not wanting to care, so why is she being critical when that reinforces his attitude towards you, and increases the pressure on her to care?
Carer’s humour warning: I told my wife yesterday that I had a favourite child. Apparently it’s supposed to be one of ours. 
Book a family holiday, for a week, there are lots of self catering offers at this time of the year.
Then maybe dad and sister will realise that they need extra support.
If dad has made it clear he doesn’t want you, then he and sister can find someone different.
You did your bit!