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Achieved spatial resolution of the PET systems is often limited by the parallax error due to the lack of information about the Depth of Interaction (DoI) inside the crystal of the incoming 511 keV annihilation photons. The smaller the diameter of the PET ring and the thicker the scintillator are, the more this error affects imaging performance. In this work, a DoI calculator suitable for monolithic scintillation crystals and based on the shape of the scintillation light distribution at the photodetector surface has been proposed. To test the estimator performance, a test PET module with a 50 \times 50 \times 20\; \mm monolithic LYSO crystal coupled to a 12 \times 12 SiPM array has been employed. In addition, for calibration and validation of the method, Geant4 simulations have been also used. The key result of the application of the proposed DoI estimator is obtaining a continuous DoI estimation with an average DoI resolution of about 5 mm in the 20 mm-thick crystal. Benefiting from the DoI estimation capabilities of the method, it has been also possible to achieve additional important goals, first of all reducing the parallax error. First, because the scintillation light collection varies as a function of the 3D position of the interaction of the annihilation photon inside the crystal, a method to correct this response variation via a proper 3D look-up-table is proposed. This has led to an improvement of about 35% in energy resolution. Moreover, a DoI-dependent position algorithm has been proposed, allowing an improvement of both planar (X-Y) position linearity and planar spatial resolution. This algorithm is specifically developed for the rows/columns multi-channel readout logic, that reduces the number of independent channels from {\rm N} \times {\rm N} to {\rm N} + {\rm N}$, where N is the number of SiPM photodetection elements (12 in our case) in each row and column. This development was performed in the framework of the MindView PET/MRI brain imaging project. [ABSTRACT FROM AUTHOR]