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 ψð3770Þ resonance. The relative strong-phase parameters between D 0 andD 0 in the decays D 0 → K 0 S;L π þ π − are studied using 2.93 fb −1 of e þ 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 → K 0 S;L π þ π − decays. Using these parameters, the associated uncertainty on the Cabibbo-Kobayashi-Maskawa angle γ=ϕ 3 is expected to be between 0.7°and 1.2°for an analysis using the decay B AE → DK AE , D → K 0 S π þ π − , where D represents a superposition of D 0 andD 0 states. This is a factor of 3 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.