We report the most precise measurements to date of the strong-phase parameters between D^{0} and D[over ¯]^{0} decays to K_{S,L}^{0}π^{+}π^{-} using a sample of 2.93 fb^{-1} of e^{+}e^{-} annihilation data collected at a center-of-mass energy of 3.773 GeV with the BESIII detector at the BEPCII collider. Our results provide the key inputs for a binned model-independent determination of the Cabibbo-Kobayashi-Maskawa angle γ/ϕ_{3} with B decays. Using our results, the decay model sensitivity to the γ/ϕ_{3} measurement is expected to be between 0.7° and 1.2°, approximately a factor of three smaller than that achievable with previous measurements, based on the studies of the simulated data. The improved precision of this work ensures that measurements of γ/ϕ_{3} will not be limited by knowledge of strong phases for the next decade. Furthermore, our results provide critical input for other flavor-physics investigations, including charm mixing, other measurements of CP violation, and the measurement of strong-phase parameters for other D-decay modes.