We present a deterministic parallel multilevel algorithm for balanced hypergraph partitioning that matches the state of the art for non-deterministic algorithms. Deterministic parallel algorithms produce the same result in each invocation, which is crucial for reproducibility. Moreover, determinism is highly desirable in application areas such as VLSI design. While there has been tremendous progress in parallel hypergraph partitioning algorithms recently, deterministic counterparts for high-quality local search techniques are missing. Consequently, solution quality is severely lacking in comparison to the non-deterministic algorithms. In this work we close this gap. First, we present a generalization of the recently proposed Jet refinement algorithm. While Jet is naturally amenable to determinism, significant changes are necessary to achieve competitive performance on hypergraphs. We also propose an improved deterministic rebalancing algorithm for Jet. Moreover, we consider the powerful but slower flow-based refinement and introduce a scheme that enables deterministic results while building upon a non-deterministic maximum flow algorithm. As demonstrated in our thorough experimental evaluation, this results in the first deterministic parallel partitioner that is competitive to the highest quality solvers. With Jet refinement, we match or exceed the quality of Mt-KaHyPar's non-deterministic default configuration while being only 15\% slower on average. We observe self-relative speedups of up to 55 on 64 cores with a 22.5$\times$ average speedup. Our deterministic flow-based refinement exceeds the quality of the non-deterministic variant by roughly 1\% on average but requires 31\% more running time.