saving PID of spawned process within a Makefile

I don't understand where you're getting stuck. What's wrong with

start:
    cd bin && { python server.py & echo $$! > server.PID; }

?

You can also make the pidfile a target and dependency:

start: server.PID

server.PID:
    cd bin && { python server.py & echo $$! > $@; }

stop: server.PID
    kill `cat $<` && rm $<

.PHONY: start stop