We derive a new parallel-in-time approach for solving large-scale optimization problems constrained by time-dependent partial differential equations arising from fluid dynamics. The solver involves the use of a block circulant approximation of the original matrices, enabling parallelization-in-time via the use of fast Fourier transforms, and we devise bespoke matrix approximations which may be applied within this framework. These make use of permutations, saddle-point approximations, commutator arguments, as well as inner solvers such as the Uzawa method, Chebyshev semi-iteration, and multigrid. Theoretical results underpin our strategy of applying a block circulant strategy, and numerical experiments demonstrate the effectiveness and robustness of our approach on Stokes and Oseen problems. Noteably, satisfying results for the strong and weak scaling of our methods are provided within a fully parallel architecture.

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