This paper describes a method of parallelisation of the popular Nelder-Mead simplex optimisation algorithms that can lead to enhanced performance on parallel and distributed computing resources. Reducing sets of simplex vertices are used to derive search directions generally closely aligned with the local gradient. When tested on a range of problems drawn from real-world applications in science and engineering, this reducing set concurrent simplex (RSCS) variant of the Nelder-Mead algorithm compared favourably with the original algorithm, and also with the inherently parallel multidirectional search algorithm (MDS). All algorithms were implemented and tested in a general-purpose, grid-enabled optimisation toolset.