I would always use "living room". "Parlour" sounds very old-fashioned, and you're unlikely to see it unless you're reading novels written in the early 1900s.
I've never heard of a "sitting room" or a "drawing room"; but, according to Google, a "sitting room" is just the British word for a "living room".
I found this website explaining the difference between a drawing room and a living room:
According to the above website, a drawing room is very similar to a living room, except its sole purpose is to receive guests (whereas a living room can do this, as well as be used for recreation).
In the house we grew up in, we had two living rooms. One had a TV and a ping pong table and we used it all the time. The other was really only used when we had people over. According to the above website, the latter room was actually a drawing room.