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We construct a parallel and explicit finite-element time-domain (FETD) algorithm for Maxwell's equations in simplicial meshes based on a mixed E- B discretization and a sparse approximation for the inverse mass matrix. The sparsity pattern of the approximate inverse is obtained from edge adjacency information, which is naturally encoded by the sparsity pattern of successive powers of the mass matrix. Each column of the approximate inverse is computed independently, allowing for different processors to be used with no communication costs and hence linear (ideal) speedup in parallel processors. The convergence of the approximate inverse matrix to the actual inverse (full) matrix is investigated numerically and shown to exhibit exponential convergence versus the density of the approximate inverse matrix. The resulting FETD time-stepping is explicit is the sense that it does not require a linear solve at every time step, akin to the finite-difference time-domain (FDTD) method.