I don't want to give my father the last of my savings so he can spend it on my grandfather's care
Doesn't your father have any of his own money?
He does but he doesn't want to spend it, he wants money from me.
It's nice to help family when it's a mutual situation. Your father seems ungrateful, though, so I can see why you're hesitant.
My father is cheap and he never wants to spend any of his own money.
Explain to your father that you don't have very much money to contribute towards your grandfather's care. He shouldn't expect you to have to contribute towards his care. Just be honest and tell him you have very little money in savings that you can contribute.