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We explore the problem of sharing data that pertains to individuals with anonymity guarantees, where each user requires a desired level of privacy. We propose the first shared-memory as well as distributed memory parallel algorithms for the k-anonimity problem that achieves this goal, and produces high quality anonymized datasets. The new algorithm is based on an optimization procedure that iteratively computes weights on the edges of a dissimilarity matrix, and at each iteration computes a minimum weighted b-edgecover in the graph. We describe how a 2-approximation algorithm for computing the b-edgecover can be used to solve the adaptive anonymity problem in parallel. We are able to solve adaptive anonymity problems with hundreds of thousands of instances and hundreds of features on a supercomputer in under five minutes. Our algorithm scales up to 8000 cores on a distributed memory supercomputer, while also providing good speedups on shared memory multiprocessors. On smaller problems where an a Belief Propagation algorithm is feasible, our algorithm is two orders of magnitude faster.