Why is killing its host usually disadvantageous for a parasite?

It leaves the parasite stranded in the host making it tough to survive!

But also nature can take it's natural course and solve the problem by

- - You Guessed It - - Mutation, adapting to the problem. By mutation

the parasite may be able to make itself air-borne, spreading from host

to host!

Sometimes the parasite might live off of the host.