The Single Source Shortest Path (SSSP) problem is a classic graph theory problem that arises frequently in various practical scenarios; hence, many parallel algorithms have been developed to solve it. However, these algorithms operate on static graphs, whereas many real-world problems are best modeled as dynamic networks, where the structure of the network changes with time. This gap between the dynamic graph modeling and the assumed static graph model in the conventional SSSP algorithms motivates this work. We present a novel parallel algorithmic framework for updating the SSSP in large-scale dynamic networks and implement it on the shared-memory and GPU platforms. The basic idea is to identify the portion of the network affected by the changes and update the information in a rooted tree data structure that stores the edges of the network that are most relevant to the analysis. Extensive experimental evaluations on real-world and synthetic networks demonstrate that our proposed parallel updating algorithm is scalable and, in most cases, requires significantly less execution time than the state-of-the-art recomputing-from-scratch algorithms.