I was never the proverbial daddy's little girl but we got along well when I was younger. He never laughs and talks with me like he does with the respite caregiver. He does, however, fight me and belittle me. I know I get crabby sometimes but I'm with him all day and the caregiver is only there for a few hours.
Also, it is a reassuring thing that you have respite care from a person who genuinely does care about her work, and has deep-rooted reasons for it.
You don't have to accept put-downs without comment. Every time he comes out with one, tell him he's doing it in whatever formula of words suits you: "don't say that, Dad, it's hurtful and unfair," for example.
The fighting you is harder: all you can do is check that any given battle is worth the trouble it causes. Discard ruthlessly any you don't really need to have.
I think that is one reason why it is so good for elders to be around people of their own age group. I noticed my Dad is a lot happier since he moved into Independent Living, but I realize that not everyone is able to make such a move.