Let us consider a regular expression r of length m and a text string T of length n over an alphabet Σ. Then, the RE minimal substring search problem is to find all minimal substrings of T matching r. Yamamoto proposed O(mn) time and O(m) space algorithm using a Thompson automaton. In this paper, we improve Yamamoto's algorithm by introducing parallelism. The proposed algorithm runs in O(mn) time in the worst case and in O(mn/p) time in the best case, where p denotes the number of processors. Besides, we show a parameter related to the parallel time of the proposed algorithm. We evaluate the algorithm experimentally.