We informally discuss several issues related to the parallel execution of logic programming systems and concurrent logic programming systems, and their generalization to constraint programming. We propose a new view of these systems, based on a particular definition of parallelism. We argue that, under this view, a large number of the actual systems and models can be explained through the application, at different levels of granularity, of only a few basic principles: determinism, non-failure, independence (also referred to as stability), granularity, etc. Also, and based on the convergence of concepts that this view brings, we sketch a model for the implementation of several parallel constraint logic programming source languages and models based on a common, generic abstract machine and an intermediate kernel language.