Crucial inputs for a variety of $CP$-violation studies can be determined through the analysis of pairs of quantum-entangled neutral $D$ mesons, which are produced in the decay of the $psi(3770)$ resonance. The relative strong-phase parameters between $D^0$ and $bar{D}^0$ in the decays $D^0rightarrow K^0_{S,L}pi^+pi^-$ are studied using 2.93~${rm fb}^{-1}$ of $e^+e^-$ annihilation data delivered by the BEPCII collider and collected by the BESIII detector at a center-of-mass energy of 3.773 GeV. Results are presented in regions of the phase space of the decay. These are the most precise measurements to date of the strong-phase parameters in $D to K_{S,L}^0pi^+pi^-$ decays. Using these parameters, the associated uncertainty on the Cabibbo-Kobayashi-Maskawa angle $gamma/phi_3$ is expected to be between $0.7^circ$ and $1.2^circ$, for an analysis using the decay $B^{pm}rightarrow DK^{pm}$, $Drightarrow K^0_Spi^+pi^-$, where $D$ represents a superposition of $D^0$ and $bar{D^0}$ states. This is a factor of three smaller than that achievable with previous measurements. Furthermore, these results provide valuable input for charm-mixing studies, other measurements of $CP$ violation, and the measurement of strong-phase parameters for other $D$-decay modes.
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