The Mathisson-Papapetrou-Dixon equations for a massive spinning test particle in plane gravitational waves are analysed and explicit solutions constructed in terms of solutions of certain linear ordinary differential equations. For harmonic waves this system reduces to a single equation of Mathieu-Hill type. In this case spinning particles may exhibit parametric excitation by gravitational fields. For a spinning test particle scattered by a gravitational wave pulse, the final energy-momentum of the particle may be related to the width, height, polarization of the wave and spin orientation of the particle.