Yes, piezoelectric pressure sensors do exist.
Of course. But do piezoelectric sensors for static pressure exist?
Yes and no, piezoelectric sensors are generally only good for measuring dynamic pressure changes but I've seen sensors that claim to be piezoelectric and capable of measuring static pressure, I seem to remember it used the principle of a SAW filter, excited by an AC signal and the resonant frequency changes as the piezoelectric element is stressed but it's been a long time since I designed in pressure sensors* so I'm not sure what the current state of the art is (or if it was just marketing wank, they were for pressures far in excess of my measuring requirements)
To answer the OP's question, piezo resistive sensors generally rely on the deflection of a sensing membrane, the 'stretch' in the sensor changes the balance of a bridge and outputs a differential voltage proportional to the deflection, google wheatstone bridge or one of the Honeywell/Sensym datasheets and application notes, they had some great ones that explained it.
* We used clamp on piezoelectric sensors for measuring and monitoring diesel fuel injector pulses as well as other pressure sensors for fuel pressure, differential air pressure and various other automotive applications, I have probably forgotten more than I remember about them now though.