Could modern unmodified Airbus aircraft be used for zero-G flight? ESA operates a mostly-stock A310 for their zero-G science flights, which doesn't have the flight envelope protection of more modern planes. Even then, a third pilot is required to monitor warnings to sort out the ones normally encountered in the parabolic maneuvers from unusual ones. If this aircraft were replaced with a modern Airbus (newer than A300/A310), would it be possible to fly these parabolic maneuvers without going into alternate law or doing other modifications to the control system?
According to this source and this source, the relevant A320 control laws (CLAWS) limitations are:


Normal load factor limitation in clean configuration down to -1g
Nose down pitch attitude down to -15deg
High Speed Protection (HSP) to prevent exceeding VD/MD


The problematic one would be the pitch attitude limiting. Since the pitch attitude will keep decreasing at zero G, the limiting function will pull the aircraft up and increase the load factor after some point. This could limit the time in zero G below the requirement. 

Given the constraints and objective, more analysis would be needed to ascertain whether aircraft in Normal Mode is acceptable.