Consider the family of rational mapsFd = {z→ fw(z) =1+ : w ∈ C\{0}} (d ∈ N, d ≥ 2)and the hyperbolic component A₁ = {w : fw has an attracting fixed point}. We prove that if w₀ ∈ ∂A₁ 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 A₁ at the point w₀; which contains w-values for which fw has an attracting periodic cycle of period q.