Despite its efficiency in conventional multiple-input multiple-output (MIMO) wireless systems, quadrature spatial modulation (QSM) becomes less efficient in massive MIMO systems since it does not adapt to the number of antennas but always uses one or two out of them. To adopt QSM in massive MIMO systems, a parallel quadrature spatial modulation (PQSM) scheme is proposed in this paper. In PQSM, the transmit (Tx) antennas are divided equally into P > 1 groups, and the bit sequence to be transmitted during a time slot is divided into P+1 parts. Then, the first part is applied to map an M-QAM complex constellation symbol while the remaining P parts of the bitstream are used to perform P QSMs in parallel. By allowing a tradeoff between the spatial modulation order and signal constellation order, PQSM enables lower bit error rate (BER) with no loss of spectral efficiency compared with QSM. For a fixed signal constellation, PQSM yields higher spectral efficiency than QSM since more selected antenna indices can carry more data bits. The algorithm pertaining to the proposed scheme is designed, and an upper bound on the average bit error rate (ABER) is derived. Moreover, to minimize the ABER, an algorithm is developed to optimize the number of Tx antenna groups and the signal constellation order. Monte-Carlo simulation results demonstrate the superiority of PQSM over generalized SM and QSM, as well as the effectiveness of the developed performance analysis.