We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous $C^{p-1}$ basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous $C^{p-1}$ IGA spaces by introducing low-continuity $C^{0}$ 0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA vs IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of $p^{2}$ when considering rIGA discretizations rather than highly-continuous IGA spaces.