my room is very cold in the winter, could my bird fluff up a lot because of that?
Yes; if they are cold they will often fluff up so that their feathers can trap a layer of warmer air around their bodies.
Pet parakeets really should not be kept anyplace that is not at least 68 degrees Fahrenheit, and 70 F is safer. (They can survive lower temps in the wild, but in the wild they can go find shelter or huddle with their flock.)
If you can't keep your room that warm, you should provide your birds a shelter inside the cage where they can huddle if they need to. You could make a little tent and/or a corner shelter out of some scrap fabric. Helwa was making some last month, I think - let me see if I can find the thread.
If your room is REALLY cold, like below 60 F most of the day, then you ought to find a warmer location for the cage. Your birds are not likely to stay healthy very long if they are not able to be warm.