Dene,
Not an Airbus Pilot myself (i fly the 737-800) but have enough mates around the place who are and in answer to your question they do. Airbus family of aircraft (319,320,321) carry a common Type Rating and so you can operate all types during a working week.
Also because the Flight Decks and FBW system is fairly common throught all the Airbus aeroplanes it is not uncommon for a Pilot in an airline that operates the narrow body family (319,320,321) and the the wide body (eg. 330) to fly both types as well. There is a course required to be taken and passed in order to do this and a qualified Airbus driver will be able to provide you much more information on how this works but I believe it's called Cross Crew Qualification (CCQ).
It's a bit like the 737. My Type Rating allows me to fly all the 737s from the 300-900, although there is a differences course for those guys moving from the 300/400/500 to the NG (600/700/800/900) and vice versa.
Hope that helps and I haven't talked too much rubbish!!