<p>This paper proposes a high-accuracy algorithm for time fractional differential equations with Caputo-Hadamard derivative. It is well known that the nonlocality inherent in fractional derivatives imposes a significant computational burden. To mitigate this issue, a spectral collocation method is adopted to preliminarily enhance computational efficiency. The resulting numerical scheme is further extended to a multi-domain framework, which enables the parallel implementation. Building upon this, a parallel algorithm is developed for the Caputo-Hadamard type initial value problem. Stability characteristics of the proposed multi-domain spectral collocation scheme is numerically discussed. Numerical examples illustrate the feasibility, stability, and convergence of the multi-domain spectral collocation method in solving both time fractional and time-space fractional differential equations with temporal Caputo-Hadamard derivative. The comparison between the proposed scheme in parallel and in sequence demonstrates the high-efficiency of the parallel multi-domain spectral collocation algorithm.</p>