If I Say " My Room Has Become A Game Parlour " Am I Grammatically Incorrect?

No, your sentence has everything it needs to be grammatically correct. It has a subject (my room) and a complete verb (has become) which agrees with the verb.

A "game parlour" may be an odd phrase, but if you mean a special room, normally for entertaining visitors, which is now being used only for playing games in, then that is a perfectly good sentence. It only sounds odd because the usual phrase is "parlour game" meaning an old-fashioned game such as charades, which were originally played at tea-parties etc (in the parlour.)