We introduce and study a parametrized analogue of the directed topological complexity, originally developed by Goubault, Farber, and Sagnier. We establish the fibrewise basic dihomotopy invariance of directed parametrized topological complexity and explore its relationship with the parametrized topological complexity. In addition, we introduce the concept of the directed Lusternik$-$Schnirelmann (LS) category, prove its basic dihomotopy invariance, and investigate its connections with both directed topological complexity and directed parametrized topological complexity. As an application, we show that the directed LS category of the directed spheres is equal to two.
Comment: 14 pages, comments are welcome!