A component-based methodology to derive parallel stiff Ordinary Differential Equation (ODE) solvers for multicomputers is presented. The methodology allows the exploitation of the multilevel parallelism of this kind of numerical algorithms and the particular structure of ODE systems by using parallel linear algebra modules. The approach furthers the reusability of the design specifications and a clear structuring of the derivation process. Two types of components are defined to enable the separate treatment of different aspects during the derivation of a parallel stiff ODE solver. The approach has been applied to the implementation of an advanced numerical stiff ODE solver on a PC cluster. Following the approach, the parallel numerical scheme has been optimized and adapted to the solution of two modelling problems which involve stiff ODE systems with dense and narrow banded structures respectively. Numerical experiments have been performed to compare the solver with the state-of-the-art sequential stiff ODE solver. The results show that the parallel solver performs specially well with dense ODE systems and reasonably well with narrow banded systems.