Influence of a magnetic field on dislocation paths in crystals with paramagnetic obstacles is studied. The model incorporates, as the most important aspect, a dependence of the energy of dislocation-obstacle bonds on their spin multiplicity. An increase of the crystal plasticity in a magnetic field is explained by an increased population of high spin states with lower binding energy. The dependence of the average dislocation path length on the magnetic field accounting also for the hyperfine interaction between electron and nucleus spins is obtained. The theory agrees well with the available experimental data in a wide range of magnetic fields. The hyperfine interaction is shown to play a crucial role, especially at relatively low magnetic fields. The possible dependence of plastic properties of crystals on the value of nuclear spin may lead to a magnetic isotope effect in plasticity. The hyperfine interaction may effectively suppress the contribution of paramagnetic obstacles in the magnetoplastic effect if the external magnetic field is smaller than a certain value that is specific for each particular type of the obstacle.