This paper presents a topology-driven parallel trajectory optimization framework, T-MPC, for dynamic environments. The approach plans multiple distinct evasive trajectories to enhance robot behavior and efficiency. A global planner iteratively generates trajectories in distinct homotopy classes, which are then optimized by parallel local planners. Each local planner is constrained to a specific homotopy class, allowing for different evasive maneuvers. The robot executes the trajectory with the lowest cost in a receding horizon manner. The method is validated on a mobile robot navigating among pedestrians, showing faster and safer trajectories than existing planners. The key contributions include a planning framework that optimizes trajectories in multiple homotopy classes in parallel, and a fast guidance planner that computes homotopy distinct trajectories through dynamic collision-free space towards multiple goals. The framework is validated in simulation and real-world scenarios, demonstrating improved performance over existing methods. The approach is implemented in C++/ROS and will be released open source. The results show that T-MPC outperforms baselines in terms of task duration, safety, and consistency, particularly in crowded environments. The method is robust to changes in the number of trajectories and consistency parameter, with optimal performance achieved at $ c_i = 0.75 $. The framework is shown to be effective in dynamic environments with multiple obstacles and dynamic obstacles.This paper presents a topology-driven parallel trajectory optimization framework, T-MPC, for dynamic environments. The approach plans multiple distinct evasive trajectories to enhance robot behavior and efficiency. A global planner iteratively generates trajectories in distinct homotopy classes, which are then optimized by parallel local planners. Each local planner is constrained to a specific homotopy class, allowing for different evasive maneuvers. The robot executes the trajectory with the lowest cost in a receding horizon manner. The method is validated on a mobile robot navigating among pedestrians, showing faster and safer trajectories than existing planners. The key contributions include a planning framework that optimizes trajectories in multiple homotopy classes in parallel, and a fast guidance planner that computes homotopy distinct trajectories through dynamic collision-free space towards multiple goals. The framework is validated in simulation and real-world scenarios, demonstrating improved performance over existing methods. The approach is implemented in C++/ROS and will be released open source. The results show that T-MPC outperforms baselines in terms of task duration, safety, and consistency, particularly in crowded environments. The method is robust to changes in the number of trajectories and consistency parameter, with optimal performance achieved at $ c_i = 0.75 $. The framework is shown to be effective in dynamic environments with multiple obstacles and dynamic obstacles.