We consider a global variable consensus ADMM algorithm for solving large-scale PDE parameter estimation problems asynchronously and in parallel. To this end, we partition the data and distribute the resulting subproblems among the available workers. Since each subproblem can be associated with different forward models and right-hand-sides, this provides ample options for tailoring the method to different applications including multi-source and multi-physics PDE parameter estimation problems. We also consider an asynchronous variant of consensus ADMM to reduce communication and latency. Our key contribution is a novel weighting scheme that empirically increases the progress made in early iterations of the consensus ADMM scheme and is attractive when using a large number of subproblems. This makes consensus ADMM competitive for solving PDE parameter estimation, which incurs immense costs per iteration. The weights in our scheme are related to the uncertainty associated with the solutions of each subproblem. We exemplarily show that the weighting scheme combined with the asynchronous implementation improves the time-to-solution for a 3D single-physics and multiphysics PDE parameter estimation problems.