Have a look at Definition of the sandbox for Windows Web Apps. If you're killing the parent process, you're killing the app's main web process. This process is used to spawn the lifetime of the app.
I wouldn't advise continuing killing the main process unless absolutely necessary. I also wouldn't advise running an exe in this manner. You didn't elaborate on the reason for doing this, but if you need to run exe then I would suggest running it as a web job instead.