The resilience of power systems is attracting extensive attention in recent years and needs to be further enhanced in the future, as potential threats from severe events such as extreme weather, geomagnetic storm, as well as extended fuel disruption, which are not easy to be quantified, predicted, or anticipated, are still challenging the modern power industry. To increase the resilience, proper operational planning considering potential impacts of severe events could effectively enable power systems to prepare for, operate through, and recover from those events and mitigate their negative economic, social, and humanitarian consequences by fully deploying existing system resources and operational measures. In this dissertation, operational planning problems in the bulk power system considering potential threats from severe events are focused, including the co-optimization of security-constrained unit commitment and transmission switching with consideration of transmission line outages probably caused by severe weather events, the security-constrained optimal power flow under potential impacts from geomagnetic storms, and the optimal operational planning to prevent electricity-natural gas systems from possible risks of natural gas supply disruptions. Notice that systematic, comprehensive, and consistent operational strategies should be conducted across the entire system to achieve superior resilience enhancement solution, which, along with increased size and complexity of modern energy systems, makes the proposed operational planning problems mathematically large-size and computationally complex optimization problems, and practically difficult to solve, especially when comprehensive operational measures and resourceful components are incorporated. In order to tackle such a challenge, the parallel optimization based approaches are developed in the proposed research, which fully decompose an originally large and complex problem into multiple independent small subproblems, simultaneously solve them in a fully parallel manner on scalable multiple-core computing platforms, and iteratively coordinate their results by using mathematical programming methods to achieve optimal solutions that satisfy engineering requirements of power system operations in practice. As a result, by efficiently solving optimal operational planning problems of large-scale power systems, their secure and economic operations in the presence of severe events like hurricanes, geomagnetic storms, and natural gas supply disruptions can be ensured, which indicates the resilience of power systems is effectively enhanced.