This paper describes ongoing research into the development of new computational algorithms and software which combine parallelism, space and time adaptivity, implicit stiff solvers, and multigrid methods. Such software is necessary in order to be able to attempt three-dimensional simulations of fundamental solidification phenomena at physically relevant parameter values. The research builds upon our prior work which successfully combined implicit stiff solvers with adaptivity and multigrid in order to simulate a class of two-dimensional solidification problems at physically realistic parameter values for the first time. The extension to three-dimensions will allow a much larger range of problems to be studied, however it requires a massive increase in computational resources which can only be delivered by the successful inclusion of efficient parallel algorithms, whilst also maintaining the use of both adaptivity and optimal multilevel solvers.