Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a subset of atoms; the second assigns each a subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently -- those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 10,000,000 atoms on three parallel supercomputers, the nCUBE 2, Intel iPSC/860, and Intel Delta. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and the Intel Delta performs about 30 times faster than a single Y-MP processor and 12 times faster than a single C90 processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.