In this work we propose a generalized joint sparsity regularization prior and reconstruction framework for the syner-gistic reconstruction of PET and undersampled sensitivity en-coded (SENSE) MRI data with the aim of improving image quality beyond that obtained through conventional independent recon-structions. The proposed prior improves upon the joint total vari-ation (TV) using a non-convex potential function that assigns a rel-atively lower penalty for the PET and MR gradients whose mag-nitudes are jointly large, thus permitting the preservation and for-mation of common boundaries irrespective of their relative orien-tation. The alternating direction method of multipliers (ADMM) optimization framework was exploited for the joint PET-MR im-age reconstruction. In this framework, the joint maximum a pos-teriori objective function was effectively optimized by alternating between well-established regularized PET and MR image recon-structions. Moreover, the dependency of the joint prior on the PET and MR signal intensities was addressed by a novel alternating scaling of the distribution of the gradient vectors. The proposed prior was compared with the separate TV and joint TV regulari-zation methods using extensive simulation and real clinical data. In addition, the proposed joint prior was compared to the recently proposed linear parallel level sets (PLS) method using a bench-mark simulation dataset. Our simulation and clinical data results demonstrated the improved quality of the synergistically recon-structed PET-MR images compared to unregularized and conven-tional separately regularized methods. It was also found that the proposed prior can outperform both joint TV and linear PLS reg-ularization methods in assisting edge preservation and recovery of details which are otherwise impaired by noise and aliasing arti-facts. In conclusion, the proposed joint sparsity regularization within the presented ADMM reconstruction framework is a prom-ising technique, nonetheless our clinical results showed that the clinical applicability of joint reconstruction might be limited in current PET-MR scanners, mainly due to the lower resolution of PET images.