Piezoelectric energy harvesters (PEH) require an electronic circuit to rectify their AC voltage in order to supply wireless sensors. In the literature, it has been shown that such electronic circuit can be tuned and optimized in order to enhance the bandwidth and power of linear PEH. However, to date, few works have explored the electrical optimization of non-linear bistable PEH. The present work reports the optimization of the harvested power and bandwidth of bistable PEH by mean of a non-linear energy extraction circuit: the parallel synchronized switch harvesting on inductor (P-SSHI). A simple model of the bistable PEH connected to the P-SSHI circuit has been derived and the closed-form expression of the harvested power has been obtained. The experimental tests, conducted on a custom-made bistable PEH showed that the results are in good agreement with the model, proving its validity. Based on the model, it is shown that the P-SSHI circuit allows to increase the bandwidth (1.6 times) and the harvested power (up to 7.5 times) compared to the standard energy harvesting (SEH) circuit.