A measurement of charm mixing and $CP$-violating parameters is reported, using $\overline{B}\ensuremath{\rightarrow}{D}^{0}(\ensuremath{\rightarrow}{K}_{\mathrm{S}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})\phantom{\rule{0ex}{0ex}}{\ensuremath{\mu}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}X$ decays reconstructed in proton-proton collisions collected by the LHCb experiment during the years 2016 to 2018, corresponding to an integrated luminosity of $5.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The measured mixing and $CP$-violating parameters are ${x}_{CP}=[4.29\ifmmode\pm\else\textpm\fi{}1.48(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.26(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, ${y}_{CP}=[12.61\ifmmode\pm\else\textpm\fi{}3.12(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}0.83(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, $\mathrm{\ensuremath{\Delta}}x=[\ensuremath{-}0.77\ifmmode\pm\else\textpm\fi{}0.93(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.28(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, $\mathrm{\ensuremath{\Delta}}y=[3.01\ifmmode\pm\else\textpm\fi{}1.92(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}0.26(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. The results are complementary to and consistent with previous measurements. A combination with the recent LHCb analysis of ${D}^{*+}\ensuremath{\rightarrow}{D}^{0}(\ensuremath{\rightarrow}{K}_{\mathrm{S}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}){\ensuremath{\pi}}^{+}$ decays is reported.