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 ψ(3770) resonance. The relative strong-phase parameters between D0 and D¯0 in the decays D0→KS,L0π+π- are studied using 2.93 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→KS,L0π+π- decays. Using these parameters, the associated uncertainty on the Cabibbo-Kobayashi-Maskawa angle γ/φ3 is expected to be between 0.7° and 1.2° for an analysis using the decay B±→DK±, D→KS0π+π-, where D represents a superposition of D0 and D¯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.