It is known that the semi-discrete ordinary differential equation (ODE) system resulting from spatial discretization of a parabolic partial differential equation, for instance, the heat equation, is highly stiff. Therefore numerical methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are preferred to solve the ODE system. However those methods are usually computationally expensive, especially for nonlinear problems. Rosenbrock method, a special subclass of implicit Runge-Kutta method, is efficient since it is iteration-free for nonlinear problem, but suffers order reduction, when it is applied to non-linear parabolic problems. In this paper we constructed a fourth-order Rosenbrock method to solve the semi-linear parabolic partial differential equation in 1-D with Dirichlet and Neumann boundary conditions. It has been shown that the Rosenbrock method is strongly A-stable hence is suitable for the stiff ODE system obtained from compact finite difference discretization of the reaction-diffusion equation. We have also shown that the new method is free of order reduction when it is applied to nonlinear parabolic problem. Several numerical examples have been solved to demonstrate the efficiency, stability and accuracy of the numerical method.