The inverse problem that underlies Magnetic Resonance Elastography (MRE) is sensitive to the measurement data and the quality of the results of this tissue elasticity imaging process can be influenced both directly and indirectly by measurement noise. In this work, we apply a coupled adjoint field formulation of the viscoelastic constitutive parameter identification problem, where the indirect influence of noise through applied boundary conditions is avoided. A well-posed formulation of the coupled field problem is obtained through conditions applied to the adjoint field, relieving the computed displacement field from kinematic errors on the boundary. The theoretical framework for this formulation via a nearly incompressible, parallel subdomain-decomposition approach is presented, along with verification and a detailed exploration of the performance of the methods via a numerical simulation study. In addition, the advantages of this novel approach are demonstrated in-vivo in the human brain, showing the ability of the method to obtain viable tissue property maps in difficult configurations, enhancing the accuracy of the method.