We present novel algorithms for parallel testing of code that takes structurally complex test inputs. The algorithms build on the Korat algorithm for constraint-based generation of structurally complex test inputs. Given an imperative predicate that specifies the desired structural constraints and a finitization that bounds the desired input size, Korat performs a systematic search to generate all test inputs (within the bounds) that satisfy the constraints. We present how to generate test inputs with a parallel search in Korat and how to execute test inputs in parallel, both off-line (when the inputs are saved on disk) and on-line (when execution immediately follows generation). The inputs that Korat generates enable bounded-exhaustive testing that checks the code under test exhaustively for all inputs within the given bounds. We also describe a novel methodology for reducing the number of equivalent inputs that Korat can generate. Our development of parallel Korat and the methodology for reducing equivalent inputs are motivated by testing an application developed at Google. The experimental results on running parallel Korat across up to 1024 machines on the Google's infrastructure show that parallel test generation and execution can achieve significant speedup, up to 543.55 times.