In this paper, convex stochastic optimization problems under different assumptions on the properties of the available stochastic subgradients are considered. It is known that if a value of the objective function is available, one can obtain, in parallel, several independent approximate solutions in terms of the objective residual expectation. Then, choosing a solution with the minimum function value, one can control the probability of large deviations of the objective residual. On the contrary, in this short paper we address the situation when the objective function value is unavailable or is too expensive to calculate. Under the “light-tail” assumption for stochastic subgradients and in the general case with a moderate probability of large deviations, it is shown that parallelization combined with averaging gives bounds for the probability of large deviations similar to those of a serial method. Thus, in these cases one can benefit from parallel computations and reduce the computational time without any loss in the solution quality.