It can, but it doesn't have to! The skin you expose by exfoliating is delicate and easy to dry out, so it's not the exfoliating itself that makes your skin feel parched, per se.
The most important thing is to give your face extra hydration after you use any kind of scrub or peel. That way you protect your lipid barrier and seal in the moisture you already have.
Also make sure that the exfoliant you're using isn't too harsh. If you're using a skin-sloughing product every day, consider using something extra mild. Pixi Glow Tonic is a great toner that also has a little acid in it for brightening and keeping mild acne away. If you're not a toner fan, there's also Aqua Glycolic Face Cream, which actually exfoliates while it hydrates—it's a longtime favorite for combating dullness and fine lines.