Numerous confluence criteria for plain term rewrite systems are known. For logically constrained rewrite system, an attractive extension of term rewriting in which rules are equipped with logical constraints, much less is known. In this paper we extend the strongly-closed and (almost) parallel-closed critical pair criteria of Huet and Toyama to the logically constraint setting. We discuss the challenges for automation and present crest, a new tool for logically constrained rewriting in which the confluence criteria are implemented, together with experimental data.