This is the third part of a trilogy on parallel solution of the linear elasticity problem. We consider the separate displacement ordering for a plain isotropic problem with full Dirichlet boundary conditions. The parallel solution methods presented in the first two parts of the trilogy are here generalised to higher order by using hierarchical finite elements. We discuss node numberings on regular grids for high degree of parallelism and even processor load as well as the problem of stability of the modified incomplete Cholesky factorisations used. Several preconditioning techniques for the conjugate gradient method are studied and compared for quadratic finite elements. Bounds for the condition numbers of the corresponding preconditioning methods are derived, and computer experiments are performed in order to confirm the theory and give recommendations on the choice of method. The parallel implementation is performed by message passing interface. Copyright © 2012 John Wiley & Sons, Ltd.