The need for parallelism in the solution of initial value problems for ordinary differential equation becomes apparent when the problem is expensive. The expensive factors are caused by either costly function evaluations, large size of the problem which contributeds to the cost of the linear algebra prosecess involved long rang interval of integration or the nature of the problem which by it self is complicated to solve. Parallelism across the method is one of the approaches dealing with such an expensive problem. Many reserachers have worked in this field to construct methods which exploit parallelism involving stage vector computation. Much of their work focused on Runge-Kutta methods and there is a lack of implmenting these methods into codes available to end users. This thesis investigates parallel approaches based on Multistep Runge-Kutta method of Radau Type. These approaches, which are extensions to the work of vander Houwen et al, are designed in such a way that the function evaluations and the process for computation of the stage vectors can be done in parallel. The stability and order of the methods are discussed both theoretically and numerically. Parallel codes are developed, and numerical tests on large dense and sparse problems are done on a SGI Power Challenge and SGI-Origin 2000.