The numerical methods used to solve the time-dependent Dirac equation on a three-dimensional Cartesian lattice are described. Efficient algorithms are required for computationally intensive studies of nonperturbative relativistic quantum dynamics. Discretization is achieved through the lattice basis-spline collocation method, in which quantum-state vectors and coordinate-space operators are expressed in terms of basis-spline functions on a spatial lattice. All numerical procedures reduce to a series of matrix-vector operations which are performed on the Intel iPSC/860 hypercube, making full use of parallelism. Solutions to the problems of limited node memory and node-to-node communication overhead inherent in using distributed-memory, multiple-instruction, multiple-data stream parallel computers are discussed.