During the Orion spacecraft’s return, at higher altitudes drogue parachutes will be used for deceleration. These parachutes are made of ribbons and have 24 gores, with 52 ribbons in each gore, where a gore is the slice of the parachute between two radial reinforcement cables extending from the parachute apex to the skirt. There are hundreds of gaps that the flow goes through, and there are also three wider gaps created by removing ribbons. Computational analysis can help reduce the number of costly drop tests in comprehensive evaluation of the parachute performance. Reliable analysis requires accurate computation of the parachute fluid-structure interaction (FSI) between the drogue and the compressible flow it is subjected to. The FSI computation is challenging because of the geometric and flow complexities and requires first creation of a starting parachute shape and flow field. This is a process that by itself is rather challenging, and that is what we are focusing on here. In our structural and fluid mechanics computations, for spatial discretization, we use isogeometric discretization with quadratic NURBS basis functions. This gives us a parachute shape that is smoother than what we get from a typical finite element discretization. In the flow analysis, we use the NURBS basis functions in the context of the compressible-flow Space-Time SUPG (ST SUPG) method. The combination of the ST framework, NURBS basis functions, and the SUPG stabilization assures superior computational accuracy.