Can you host a caisa on an undezzed piece of ice ? Seems it would make sense for rook ? And while we're at it, parasite ?

caisa, yes... for the most part (I don't have the card text infront of me, but I can't recall any that restrict whether the ice needs to be rezzed or unrezzed). parasite, no.

Parasite will stay hosted on a piece of ice and continue to accumulate virus tokens if the ice gets derezzed, but it specifically says you can only install it hosted on a rezzed piece of ice.

Caissa definitely yes: Rook on a rezzed piece of ice is less than optimal.