As computer architectures have advanced to utilize multiple computational cores in parallel, few methods have been proposed to take advantage of parallelism for solving Initial Value Problems. Motivated by a desire to improve the computational performance of high accuracy orbit propagation, this work explores the application of a parallel integration technique called Revisionist Integral Deferred Correction (RIDC), to the propagation of satellite orbits. This thesis explores the development of the theory behind spectral correction techniques and uses this foundation to develop the theoretical underpinnings of the RIDC method. This theoretical background is then used to design and develop Fixed and Adaptive-step versions of this integration method implemented in the Rust programming language. In order to facilitate an accurate comparison between RIDC and other common integration methods, Runge-Kutta and Adams-Bashforth methods were also implemented in the Rust programming language. These methods for the solution of Initial Value Problems were then tested on a one-dimensional test problem. This work used the test problem to compare the performance of the RIDC method to other common integration techniques and to explore the effects of tuning parameters. The RIDC implementation was then tested on perturbed and unperturbed versions of Keplers problem where the RIDC method achieved lower than 10^-7 km positional error and 10^-7 km/s velocity error over a 3 hour integration time for both versions of the problem. These accuracies are shown to be comparable to the accuracy achieved with a Dormand-Prince 7(8) integrator. The RIDC method was unable to outperform the high-order Dormand-Prince 7(8) integrator in terms of computational time in the numerical experiments. However, the RIDC method's performance suggests that it is a promising candidate for further study for the parallel computation of high-accuracy solutions to the propagation of satellite orbits.