This paper is concerned with the problem of output feedback control for a class of continuous-time hidden semi-Markov jump systems with time delays. Due to the limitations of the actual environment, system modes are usually undetectable, which are called hidden modes. The controller modes are described as observable modes. Emission probability is used to establish the relationship between above-mentioned two concepts. The jump parameters are governed by the hidden semi-Markov process, and the reason is that the mode information of the controller and the system is asynchronous. Besides, time delays are considered to be time-varying and dependent on the hidden mode. By employing some mathematical transformation and constructing a novel Lyapunov-Krasovskii functional, some new parameter-dependent sufficient stablization conditions can be obtained by designing an observed-mode-dependent static output-feedback controller. Finally, a practical example is provided to illustrate the effectiveness and merits of the proposed methods.