In this paper, we consider the problem of scheduling jobs on parallel identical machines, where the processing times of jobs are uncertain: only interval bounds of processing times are known. The optimality criterion of a schedule is the total completion time. In order to cope with the uncertainty, we consider the maximum regret objective and we seek a schedule that performs well under all possible instantiations of processing times. Although the deterministic version of the considered problem is solvable in polynomial time, the minmax regret version is known to be weakly NP-hard even for a single machine, and strongly NP-hard for parallel unrelated machines. In this paper, we show that the problem is strongly NP-hard also in the case of parallel identical machines.