  We study the Gevrey character of a natural parameterization of one
dimensional invariant manifolds associated to a parabolic direction of fixed
points of analytic maps, that is, a direction associated with an eigenvalue
equal to $1$. We show that, under general hypotheses, these invariant manifolds
are Gevrey with type related to some explicit constants. We provide examples of
the optimality of our results as well as some applications to celestial
mechanics, namely, the Sitnikov problem and the restricted planar three body
problem.
