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We study the existence of Fatou components on parabolic skew product maps. We focus on skew products in which each coordinate has a fixed point that is parabolic. As in the geometrically attracting case, we prove that there exists maps F that have one-dimensional disks that are mapped to a point in the Julia set of the restriction of F to an invariant one-dimensional fiber. We first prove a linearization theorem for a one-dimensional map, then for a parabolic skew product. Finally, we apply this result to construct the skew product map described above.

Parametrization of unstable manifolds and Fatou disks for parabolic skew-products