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Random graphs (or networks) have gained a significant increase of interest due to its popularity in modeling and simulating many complex real-world systems. Degree sequence is one of the most important aspects of these systems. Random graphs with a given degree sequence can capture many characteristics like dependent edges and non-binomial degree distribution that are absent in many classical random graph models such as the Erdöos-Rényi graph model. In addition, they have important applications in uniform sampling of random graphs, counting the number of graphs having the same degree sequence, as well as in string theory, random matrix theory, and matching theory. In this paper, we present an OpenMP-based shared-memory parallel algorithm for generating a random graph with a prescribed degree sequence, which achieves a speedup of 20.4 with 32 cores. We also present a comparative study of several structural properties of the random graphs generated by our algorithm with that of the real-world graphs and random graphs generated by other popular methods. One of the steps in our parallel algorithm requires checking the Erdöos-Gallai characterization, i.e., whether there exists a graph obeying the given degree sequence, in parallel. This paper presents a non-trivial parallel algorithm for checking the Erdöos-Gallai characterization, which achieves a speedup of 23 with 32 cores.