Blackbox optimization refers to the situation where no analytical knowledge about the problem is available beforehand, which is the case in a number of application fields, e.g., multi-disciplinary design, simulation optimization. In this context, the so-called Simultaneous Optimistic Optimization (SOO) algorithm is a deterministic tree-based global optimizer exposing theoretically provable performance guarantees under mild conditions. In this paper, we consider the efficient shared-memory parallelization of SOO on a high-end HPC architecture with dozens of CPU cores. We thereby propose different strategies based on eliciting the possible levels of parallelism underlying the SOO algorithm. We show that the naive approach, performing multiple evaluations of the blackbox function in parallel, does not scale with the number of cores. By contrast, we show that a parallel design based on the SOO-tree traversal is able to provide substantial improvements in terms of scalability and performance. We validate our strategies with a detailed performance analysis on a compute server with two 64-core processors, using a number of diverse benchmark functions with both increasing dimensions and number of cores.