We report a direct measurement of D{sup 0}-D{sup 0} mixing parameters through a time-dependent amplitude analysis of the Dalitz plots of D{sup 0}{yields}K{sub S}{sup 0}{pi}{sup +}{pi}{sup -} and, for the first time, D{sup 0}{yields}K{sub S}{sup 0}K{sup +}K{sup -} decays. The low-momentum pion {pi}{sub s}{sup +} in the decay D{sup *+{yields}}D{sup 0{pi}}{sub s}{sup +} identifies the flavor of the neutral D meson at its production. Using 468.5 fb{sup -1} of e{sup +}e{sup -} colliding-beam data recorded near {radical}(s)=10.6 GeV by the BABAR detector at the PEP-II asymmetric-energy collider at SLAC, we measure the mixing parameters x=1.6{+-}2.3(stat){+-}1.2(syst){+-}0.8(model)x10{sup -3}, and y=5.7{+-}2.0(stat){+-}1.3(syst){+-}0.7(model)x10{sup -3}. These results provide the best measurement to date of x and y. The knowledge of the value of x, in particular, is crucial for understanding the origin of mixing.