Recently, full-duplex (FD) communications with simultaneous transmission and reception on the same channel has been proposed. The FD receiver, however, suffers from inevitable self-interference (SI) from the much more powerful transmit signal. Analogue radio-frequency (RF) and baseband, as well as digital baseband, cancellation techniques have been proposed for suppressing the SI, but so far most of the studies have failed to take into account the inherent nonlinearities of the transmitter and receiver front-ends. To fill this gap, this article proposes a novel digital nonlinear interference cancellation technique to mitigate the power amplifier (PA) induced nonlinear SI in a FD transceiver. The technique is based on modeling the nonlinear SI channel, which is comprised of the nonlinear PA, the linear multipath SI channel, and the RF SI canceller, with a parallel Hammerstein nonlinearity. Stemming from the modeling, and appropriate parameter estimation, the known transmit data is then processed with the developed nonlinear parallel Hammerstein structure and suppressed from the receiver path at digital baseband. The results illustrate that with a given IIP3 figure for the PA, the proposed technique enables higher transmit power to be used compared to existing linear SI cancellation methods. Alternatively, for a given maximum transmit power level, a lower-quality PA (i.e., lower IIP3) can be used.