We present the first numerical model of the magnetohydrodynamical dynamo, driven by cosmic rays in the interstellar medium. The cosmic rays are produced in randomly occurring supernova remnants. Our model is described by equations of magnetohydrodynamics supplemented with the diffusion-advectionequation governing the propagation of cosmic rays. The other essential elements of the model are: vertical gravity of the disk, differential rotation and resistivity leading to reconnection of magnetic field lines. We obtain amplification of the large-scale magnetic field on a timescale of galactic rotation. The model represents a kind of fast, cosmic ray driven galactic dynamo proposed by Parker (1992). The amplification of the large-scale magnetic field results from the buoyancy of the cosmic ray component in the interstellar medium.