Is there a specific word for cutting up a chicken into parts? Like how "field dressing" means to remove organs from game, is there a term for cutting a chicken into its respective parts?
In the UK at least you can "joint" a chicken. British TV chef and cookery book writer Delia Smith gives full instructions here but other online sources also use the term. Whether it's used in the US I don't know but as John Lawler suggests "cut up" in his comment I suspect not.
You can also spatchcock or butterfly a chicken but this merely removes some of the bones and lays the bird flat for faster cooking, it doesn't cut it into pieces.
Butchering an animal does not imply that it is alive. You can butcher a dead animal. In fact that is a normal use of "butcher", being what a butcher (noun) does.
A verb for cutting up a chicken is to fabricate. My son is in culinary arts and this is what I just learned today from him. I googled "fabricate a chicken" & "chicken fabrication". Never knew it had anything to do with cutting up chicken. HOW TO: FABRICATE CHICKEN
So far mentioned: Butching, jointing, cutting up/cut up. Let me add "Break down."
See it used here: https://www.seriouseats.com/better-cook-buy-whole-chickens-cut-up-pieces