We study the performance of a parametric quadrature amplitude modulation (QAM) family, called theta-QAM, which includes other known constellations, such as square QAM (SQAM) and triangular QAM (TQAM), as special cases. The versatile structure of the theta-QAM signal constellation, which occurs from the varying angle between the signal points, results in achieving the minimum symbol error rate (SER) or bit error rate (BER), under an average power constraint. Exact analytical expressions are obtained for the SER performance of theta-QAM in additive white Gaussian Noise (AWGN) and Nakagami-m fading channels, while highly accurate approximations for the BER are presented. The theoretical study aims at providing insight into the trade-off between error performance and complexity of this parametric modulation scheme. The presented theoretical analysis is validated via extensive computer simulations.