The authors develop designs for the data parallel solution of quadratic optimization problems subject to box constraints. In particular, they consider the class of algorithms that iterate between projection steps that identify candidate active sets and conjugate gradient steps. Using the algorithms of \textit{J. H. Moré} and \textit{G. Toraldo} [ibid. 1, No. 1, 93-113 (1991; Zbl 0752.90053)] as a specific instance of these algorithms, they show how its components can be implemented efficiently on a data-paralled SIMD computer architecture. Alternative designs are developed for unstructured Hessian and for structured problems. Implementations are carried out on a Connection Machine CM-2. They are shown to be very efficient, for instance, problems with several hundred thousand variables are solved within one minute on the 8K CM-2.