The Block Recursive Inversion (BRI) algorithm calculates the inversion of large k x k block matrices with limited memory during the entire processing because it calculates one block of the inverse at a time. However, the lower memory consumption is counterbalanced by higher computational complexity. We propose a parallel BRI implementation, which also calculates one block at a time, to reduce execution time and extend its applicability by exploiting modern multi-core architectures. The proposed parallel BRI was implemented for shared memory systems in OpenMP. The results of a performance and scalability analysis for different use cases reveals opposite trends in execution time, with the proposed parallel implementation being faster for larger k. Although not weakly scalable for a fixed k, scalability tends to increase with the increase of k or, equivalently, with the reduction of memory requirements.