Slim structures are usually thin or slender, and modeling their interaction with the continuum using a single approach will significantly increase the computational cost due to the scale mismatch. To address this challenge, this paper proposes a GPU-accelerated coupled material point method (MPM) and discrete element method (DEM) framework to simulate the continuum-structure interaction. Within this framework, MPM models the mechanical behavior of a large-deformation continuum, while DEM captures the stress and deformation of the slim structures. A novel mixed MPM-DEM contact model based on the MPM background grid efficiently handles the interaction between the slim structure and the continuum. The proposed framework enables continuous integration of DEM into MPM, facilitating finer modeling of slim structures, leveraging the strengths of both methods. To evaluate the reliability of DEM for modeling slim structures and the efficacy of the MPM-DEM in capturing interactions between continuum and slim structures, five benchmark cases are examined: block sliding on an inclined plane, tensile test of geogrid, rebars in a cantilever beam, punch test of wire mesh, and debris flow impacts on a flexible net barrier. The results demonstrate significantly accurate modeling capabilities, establishing the framework as a robust tool for applications in geotechnical engineering problems.