We introduce a new algorithm to solve a regularized spatial-spectral image estimation problem. Our approach is based on the linearized alternating directions method of multipliers (LADMM), which is a variation of the popular ADMM algorithm. Although LADMM has existed for some time, it has not been very widely used in the computational imaging literature. This is in part because there are many possible ways of mapping LADMM to a specific optimization problem, and it is nontrivial to find a computationally efficient implementation out of the many competing alternatives. We believe that our proposed implementation represents the first application of LADMM to the type of optimization problem considered in this work (involving a linear-mixture forward model, spatial regularization, and nonnegativity constraints). We evaluate our algorithm in a variety of multiparametric MRI partial volume mapping scenarios (diffusion-relaxation, relaxation-relaxation, relaxometry, and fingerprinting), where we consistently observe substantial ($\sim$3$\times$-50$\times$) speed improvements. We expect this to reduce barriers to using spatially-regularized partial volume compartment mapping methods. Further, the considerable improvements we observed also suggest the potential value of considering LADMM for a broader set of computational imaging problems.