The iterative, multigroup, discrete ordinates Sn representation for the linear transport equation enjoys widespread computational use and popularity. Serial iteration schemes and numerical algorithms developed over the years provide a timely framework for parallel extension. On the Denelcor heterogeneous element processor, three parallel iteration schemes (two chaotic, one ordered) are investigated for solving the one-dimensional Sn transport equation. Concurrent inner sweeps, coupled acceleration techniques, synchronized inner-outer loops, and chaotic iteration are described and results of computations are contrasted. The multigroup representation and serial iteration methods are also reviewed. The basic iterative Sn approach lends itself to parallel tasking, portably affording an effective medium for performing transport calculations on future architectures. This analysis represents a first attempt to extend serial Sn algorithms to parallel environments and provides good baseline estimates on ease of parallel implementation, relative algorithm efficiency, comparative speedup, and some future directions. An ordered and chaotic strategy is examined, with and without concurrent rebalance and diffusion acceleration, which efficiently support high degrees of parallelization and appear to be robust and viable parallel iteration techniques. The standard inner-outer technique, presently employed in a majority of production Sn codes, is a weaker parallel iteration strategy. Modifications, extensions, and recoding effort to parallelize existing serial algorithms are also simple. Chaotic iteration, heretofore difficult to simulate on serial machines, holds promise and appears to converge faster than ordered schemes. Actual parallel speedup and efficiency are high and payoff appears substantial.