Running evolutionary algorithms in parallel is an intuitive way to speed up the process of solving large-scale multi-objective optimization problems, which have hundreds or thousands of decision variables. However, the framework of the existing multi-objective evolutionary algorithms seriously limits their parallelization. During each iteration, the environmental selection operators present in the existing framework need to collect and compare all the candidate solutions to balance the convergence and diversity, thus dividing the whole evolutionary process into a series of dependent sub-processes and resulting in frequent data transmission. To address this issue, we propose a novel parallel framework that separates the environmental selection operator from the entire evolutionary process, evidently removing the dependencies among sub-processes and reducing the data transmission. On the basis of the parallel framework, a new parallel evolutionary algorithm, namely PEA, is designed. In PEA, the convergence is achieved by a series of independent sub-populations, and the diversity is merely emphasized at the converged solutions from each subpopulation, which is helpful for avoiding that the environmental selection operator limits the parallelization of the algorithm. Moreover, a new environmental selection strategy is proposed to improve the diversity without considering the convergence. To assess the performance of the proposed PEA, we compare it with five representative multi-objective evolutionary algorithms in terms of both the convergence and diversity. The performance of the parallel framework is also analyzed by comparing with two existing parallel models. The experimental results demonstrate the superiority of the proposed parallel algorithms in terms of the convergence, diversity, and speedup.