Hexagonal boron nitride (hBN) is a prototypical high-quality two-dimensional insulator and an ideal material to study tunneling phenomena, as it can be easily integrated in vertical van der Waals devices. For spintronic devices, its potential has been demonstrated both for efficient spin injection in lateral spin valves and as a barrier in magnetic tunnel junctions (MTJs). Here we reveal the effect of point defects inevitably present in mechanically exfoliated hBN on the tunnel magnetoresistance of Co-hBN-NiFe MTJs. We observe a clear enhancement of both the conductance and magnetoresistance of the junction at well-defined bias voltages, indicating resonant tunneling through magnetic (spin-polarized) defect states. The spin polarization of the defect states is attributed to exchange coupling of a paramagnetic impurity in the few-atomic-layer thick hBN to the ferromagnetic electrodes. This is confirmed by excellent agreement with theoretical modelling. Our findings should be taken into account in analyzing tunneling processes in hBN-based magnetic devices. More generally, our study shows the potential of using atomically thin hBN barriers with defects to engineer the magnetoresistance of MTJs and to achieve spin filtering, opening the door towards exploiting the spin degree of freedom in current studies of point defects as quantum emitters.