Summary: The random-adversary technique is a general method for proving lower bounds on randomized parallel algorithms. The bounds apply to the number of communication steps, and they apply regardless of the processors' instruction sets, the lengths of messages, etc. This paper introduces the random-adversary technique and shows how it can be used to obtain lower bounds on randomized parallel algorithms for load balancing, compaction, padded sorting, and finding Hamiltonian cycles in random graphs. Using the random-adversary technique, we obtain the first lower bounds for randomized parallel algorithms which are provably faster than their deterministic counterparts (specifically, for load balancing and related problems).