  We initiate the study of parallel algorithms for fairly allocating
indivisible goods among agents with additive preferences. We give fast parallel
algorithms for various fundamental problems, such as finding a Pareto Optimal
and EF1 allocation under restricted additive valuations, finding an EF1
allocation for up to three agents, and finding an envy-free allocation with
subsidies. On the flip side, we show that fast parallel algorithms are unlikely
to exist (formally, $CC$-hard) for the problem of computing Round-Robin EF1
allocations.
