  The P2 experiment aims at high-precision measurements of the parity-violating
asymmetry in elastic electron-proton and electron-$^{12}$C scatterings with
longitudinally polarized electrons. We discuss here the sensitivity of P2 to
new physics mediated by an additional neutral gauge boson $Z'$ of a new $U(1)'$
gauge symmetry. If the charge assignment of the $U(1)'$ is chiral, i.e., left-
and right-handed fermions have different charges under $U(1)'$, additional
parity-violation is induced directly. On the other hand, if the $U(1)'$ has a
non-chiral charge assignment, additional parity-violation can be induced via
mass or kinetic $Z$-$Z'$ mixing. By comparing the P2 sensitivity to existing
constraints, we show that in both cases P2 has discovery potential over a wide
range of $Z'$ mass. In particular, for chiral models, the P2 experiment can
probe gauge couplings at the order of $10^{-5}$ when the $Z'$ boson is light,
and heavy $Z'$ bosons up to 79 (90) TeV in the proton ($^{12}$C) mode. For
non-chiral models with mass mixing, the P2 experiment is sensitive to mass
mixing angles smaller than roughly $10^{-4}$, depending on model details and
gauge coupling magnitude.
