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A block-iterative parallel decomposition method is proposed to solve general quadratic signal recovery problems under convex constraints. The proposed method proceeds by local linearizations of blocks of constraints, and it is therefore not sensitive to their analytical complexity. In addition, it naturally lends itself to implementation on parallel computing architectures due to its flexible block-iterative structure. Comparisons with existing methods are carried out, and the case of inconsistent constraints is also discussed. Numerical results are presented.