Iterative methods often provide the only means of solving large eigenvalue problems. Their block variants converge slowly but they are robust especially in the presence of multiplicities. Preconditioning is often used to improve convergence. Yet, for large matrices, the demands posed on the available computing resources are huge. Clusters of workstations and SMPs are becoming the main computational tool for many scientific and engineering groups. In these environments, high communication costs suggest coarse grain implementations of iterative methods. We have combined the block and preconditioning functionalities into a parallel implementation of a block Jacobi-Davidson method. We combine a finegrain iterative scheme with the coarse grain capability that each processor can precondition different eigenpairs. We outline the design and present some timings and convergence results on a small workstation cluster and on a SUN Enterprise. Keywords: Eigenvalue, preconditioning, block, paral..