In this paper, a parallel domain decomposition method is proposed for solving the fully-mixed Stokes-dual-permeability fluid flow model with Beavers-Joseph (BJ) interface conditions. Three Robin-type boundary conditions and a modified weak formulation are constructed to completely decouple the original problem, not only for the free flow and dual-permeability regions but also for the matrix and microfracture flow fields in the dual-porosity media. We derive the equivalence between the original problem and the decoupled systems with some suitable compatibility conditions, and also demonstrate the equivalence of two weak formulations in different Sobolev spaces. Based on the completely decoupled modified weak formulation, the mesh-independent geometric convergence rate of the proposed iterative parallel algorithm is proved rigorously with some suitable parameters. To carry out the convergence analysis of our proposed algorithm, we utilize an important but general convergence lemma for the steady-state problems. Finding the optimized Robin parameters that can accelerate the convergence of the proposed algorithm is an open problem mentioned inhere. Finally, several numerical experiments are presented to illustrate and validate the exclusive features of our proposed algorithm.