The phase field modeling of fracture is able to simulate the nucleation and the propagation of complex crack patterns. However, the relatively small internal lengths that are required usually lead to very fine meshes and high computational costs, especially for three-dimensional applications. Unstable crack propagations are regularized through an implicit dynamics framework potentially leading to a very large variation of time steps switching from a quasi-static regime to a dynamic one. This regularization technique increases even more the computational cost of the simulation. To reduce the time to solution and exploit modern supercomputers, we propose a domain decomposition framework and acceleration techniques for the phase field fracture staggered solver. The displacement subproblem and the phase field one are solved with parallel domain decomposition solvers. Dual domain decomposition methods provide low cost preconditioner well adapted to the phase field subproblem. For displacement subproblems undergoing unstable crack propagations, primal domain decomposition methods are preferred since they are less sensitive to the treatment of floating substructures. Preconditioners performances are assessed and scalability studies over academic test cases, up to 324 subdomains, are presented. Finally, the robustness of the approach is illustrated on two semi-industrial simulations.