  In this paper, we prove that fractal zeta functions of orbits of parabolic
germs of diffeomorphisms can be meromorphically extended to the whole complex
plane. We describe their set of poles (i.e. their complex dimensions) and their
principal parts which can be understood as their fractal footprint. We study
the fractal footprint of one orbit of a parabolic germ f and extract intrinsic
information about the germ f from it, in particular, its formal class.
Moreover, we relate complex dimensions to the generalized asymptotic expansion
of the tube function of orbits with oscillatory "coefficients" as well as to
the asymptotic expansion of their dynamically regularized tube function.
Interestingly, parabolic orbits provide a first example of sets that have
nontrivial Minkowski (or box) dimension and their tube function possesses
higher order oscillatory terms, however, they do not posses non-real complex
dimensions and are therefore not called fractal in the sense of Lapidus.
