  We have implemented the dynamical vertex approximation (D$\Gamma$A) in its
full parquet-based version to include spatial correlations on all length scales
and in {\sl all} scattering channels. The algorithm is applied to study the
electronic self-energies and the spectral properties of finite-size
one-dimensional Hubbard models with periodic boundary conditions (nanoscopic
Hubbard rings). From a methodological point of view, our calculations and their
comparison to the results obtained within dynamical mean-field theory, plain
parquet approximation, and the exact numerical solution, allow us to evaluate
the performance of the D$\Gamma$A algorithm in the most challenging situation
of low dimensions. From a physical perspective, our results unveil how
non-local correlations affect the spectral properties of nanoscopic systems of
various sizes in different regimes of interaction strength.
