Consider the family of rational mapsFd = {z? fw(z) =1+ : w ? C\{0}} (d ? N, d ? 2)and the hyperbolic component A? = {w : fw has an attracting fixed point}. We prove that if w? ? ?A? is a parabolic parameter with corresponding multiplier a primitive q-th root of unity, q ? 2; then there exists a hyperbolic component Wq; attached to A? at the point w?; which contains w-values for which fw has an attracting periodic cycle of period q.