A search for $C$P and $P$ violation using triple-product asymmetries is performed with $\Lambda^{0}_{b}\to pK^{-}\pi^{+}\pi^{-}$, $\Lambda^{0}_{b}\to pK^{-}K^{+}K^{-}$ and $\Xi^{0}_{b}\to pK^{-}K^{-}\pi^{+}$ decays. The data sample corresponds to integrated luminosities of 1.0fb$^{-1}$ and 2.0fb$^{-1}$, recorded with the LHCb detector at centre-of-mass energies of 7TeV and 8TeV, respectively. The $CP$- and $P$-violating asymmetries are measured both integrating over all phase space and in specific phase-space regions. No significant deviation from $CP$ or $P$ symmetry is found. The first observation of $\Lambda^{0}_{b}\to pK^{-}\chi_{c0}(1P)(\to\pi^{+}\pi^{-}, K^{+}K^{-})$ decay is also reported.