Local search is a successful approach for solving combinatorial optimization and constraint satisfaction problems. With the progressing move toward multi and many-core systems, GPUs and the quest for Exascale systems, parallelism has become mainstream as the number of cores continues to increase. New programming models are required and need to be better understood as well as data structures and algorithms. Such is the case for local search algorithms when run on hundreds or thousands of processing units. In this paper, we discuss some experiments we have been doing with Adaptive Search and present a new parallel version of it based on GPI, a recent API and programming model for the development of scalable parallel applications. Our experiments on different problems show interesting speedups and, more importantly, a deeper interpretation of the parallelization of Local Search methods.