  We consider a map $F$ of class $C^r$ with a fixed point of parabolic type
whose differential is not diagonalizable and we study the existence and
regularity of the invariant manifolds associated with the fixed point using the
parameterization method. Concretely, we show that under suitable conditions on
the coefficients of $F$, there exist invariant curves of class $C^r$ away from
the fixed point, and that they are analytic when $F$ is analytic. The
differentiability result is obtained as an application of the fiber contraction
theorem. We also provide an algorithm to compute an approximation of a
parameterization of the invariant curves and a normal form of the restricted
dynamics of $F$ on them.
