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Context. The existence of a maximum correlation angle ($θ_{max} \gtrsim 60^{\circ}$) in the two-point angular temperature correlations of cosmic microwave background (CMB) radiation, measured by WMAP and Planck, stands in sharp contrast to the prediction of standard inflationary cosmology, in which the correlations should extend across the full sky (i.e., $180^{\circ}$). The introduction of a hard lower cutoff ($k_{min}$) in the primordial power spectrum, however, leads naturally to the existence of $θ_{max}$. Among other cosmological anomalies detected in these data, an apparent dominance of odd-over-even parity multipoles has been seen in the angular power spectrum of the CMB. This feature, however, may simply be due to observational contamination in certain regions of the sky. Aims. In attempting to provide a more detailed assessment of whether this odd-over-even asymmetry is intrinsic to the CMB, we therefore proceed in this paper, first, to examine whether this odd-even parity imbalance also manifests itself in the angular correlation function and, second, to examine in detail the interplay between the presence of $θ_{max}$ and this observed anomaly. Methods. We employed several parity statistics and recalculated the angular correlation function for different values of the cutoff $k_{min}$ in order to optimize the fit to the different Planck 2018 data. Results. We find a phenomenological connection between these features in the data, concluding that both must be considered together in order to optimize the theoretical fit to the Planck 2018 data. Conclusions. This outcome is independent of whether the parity imbalance is intrinsic to the CMB, but if it is, the odd-over-even asymmetry would clearly point to the emergence of new physics.