  A search for $CP$ and $P$ violation in charmless four-body $B^{0} \to p \bar
p K^{+} \pi^{-}$ decays is performed using triple-product asymmetry
observables. It is based on proton-proton collision data collected by the LHCb
experiment at centre-of-mass energies of $7$, $8$ and $13$ TeV, corresponding
to a total integrated luminosity of $8.4$ fb$^{-1}$. The $CP$- and
$P$-violating asymmetries are measured both in the integrated phase space and
in specific regions. No evidence is seen for $CP$ violation. $P$-parity
violation is observed at a significance of 5.8 standard deviations
