I believe one of the reasons to use parchment paper over foil is due to tomatoes and tomato sauce - the acids in which will react to the aluminum, giving the food a bit of an "off" taste. In those situations it's better to use parchment when possible, or even that new foil with parchment on one side, which works really well (though it's expensive). The non-stick foil is also a good alternative since it has some sort of silicone coating on it which won't affect the tomato sauce.
I would say the down side to cooking with parchment is you need to keep an eye on it in an oven because I've seen it begin to smolder and catch fire at the edges from time to time. In most cases it will just brown a little, but at higher temps and/or longer cooking times, it will sometimes burn instead.
Personally I don't use it (parchement) all that often - mainly just for cookies, or for some "pouch" meals, where you put something like a piece of fish and some veggies and wrap them in a pouch and let them steam in their own juices in an oven.