A parametric study is presented, which employs a new anisotropic constitutive law in order to study the influence of anisotropic plasticity on the deformation field of the Asymmetric Rolling (ASR) process. A version of the facet method is presented, where an analytical yield function is restricted to the subspace of the stress and strain rate space relevant for 2D Finite Element Analysis (FEA), but can still accurately reproduce the plastic anisotropy of an underlying Crystal Plasticity (CP) model. The influence of anisotropy on the deformation field and corresponding texture evolution is examined in terms of the changes in texture component volume fractions and formation of texture gradients. It is found that a material with the anisotropy of a sharp cold-rolled aluminium alloy is more beneficial than that of a recrystallised hot-rolled aluminium alloy, and this influence of anisotropy suggests that Asymmetric Rolling (ASR) may be best carried out in the latest stages of cold rolling.