Let G be a Garside group endowed with the generating set S of non-trivial simple elements, and let H be a parabolic subgroup of G. We determine a transversal T of H in G such that each theta is an element of T is of minimal length in its right-coset, H theta, for the word length with respect to S. We show that there exists a regular language L on S boolean OR S-1 and a bijection ev: L -> T satisfying lg(U) = lg(S)(ev(U)) for all U is an element of L. From this we deduce that the coset growth series of H in G is rational. Finally, we show that G has fellow projections on H but does not have bounded projections on H.