In this paper, we investigate stochastic partial differential equations driven by multi-parameter anisotropic fractional Levy noises, including the stochastic Poisson equation, the linear heat equation, and the quasi-linear heat equation. Well-posedness of these equations under the fractional noises will be addressed. The multi-parameter anisotropic fractional Levy noise is defined as the formal derivative of the anisotropic fractional Levy random field. In doing so, there are two folds involved. First, we consider the anisotropic fractional Levy random field as the generalized functional of the path of the pure jump Levy process. Second, we build} the Skorohod integration with respect to the multi-parameter anisotropic fractional Levy noise by white noise approach.