A quark and antiquark have opposite parity. The quark is customarily taken to have positive parity. I understand this to mean that Pf = f, where f is the wavefunction of the quark and Pg = -g, where g is the wavefunction of the antiquark. Does this mean that P acting on an antiquark wavefunction flips the handedness of the particle, but won't do this for quarks? I don't think that makes sense.