We develop a data-driven method to learn Koopman-type decompositions for non-autonomous dynamical systems, including equations with static or time-varying parameters. Previous works on constructing Koopman operator dynamics were either limited to autonomous ones, or those whose dynamics are linear/bilinear in the state and parameters. In contrast, our method, which combines machine learning and Koopman operator theory, can handle general non-linear parametric dynamics, allowing greater flexibility in its application. We show theoretically the feasibility of our approach and demonstrate its performance on a variety of applications, including forward prediction problems and optimal control problems.