Based on the two-parameter Pasternak model of elastic foundation, the two-parameter model of viscoelastic foundation was derived by using fractional derivative. The equation of elastic and viscoelastic rectangular plate under the dynamic load on two-parameter viscoelastic foundation with the fractional Kelvin model was established. The equation of elastic and viscoelastic rectangular plate with four edges simply supported was solved by the Galerkin method and the segmented numerical method, the correctness of the solution was verified by the example of free vibration. The influences of the fractional order, viscosity parameter, horizontal shear parameter and modulus parameter on the displacement of the fractional Kelvin model with impact load were analyzed. The results show that the fractional derivative viscoelastic model may describe the mechanical behavior of different viscoelastic materials; the displacement response of rectangular plate appears different attenuation formation before and after the fractional order value of 0.5; the attenuation speed of the displacement response increases with the increasing of the viscosity parameter, horizontal shear parameter and modulus parameter.