We have developed a computational model to study the structural dynamics of parachute deployment using adaptive finite element analysis. The mathematical model uses a large deformation formulation of nonlinear elasticity with two dimensional membranes and one dimensional stiffeners. Constant or projected pressure loading is supported with the model, but fully coupled fluid-structure interaction is not addressed in this work. We leveraged the multiphysics package GRINS wherein we implemented large deformation membrane and cable finite element formulations. GRINS is built on the libMesh finite element library. A key aspect of libMesh is that it supports adaptive mesh refinement (AMR) on general unstructured meshes on parallel supercomputers. The novel contribution from this project is added support for AMR with mesh topologies that have conforming one- and two-dimensional manifolds as well as possible coupling to three-dimensional elements. To the best of our knowledge, this is a unique capability in open-source finite element libraries. The AMR technology is illustrated on several examples for large deformation and materially nonlinear geometries with conforming one-dimensional and two-dimensional manifolds. A final illustration is shown on a parachute geometry similar to a parachute used in the Orion space vehicle.