We present improved bounding procedures in the context of a recent global optimization algorithm based on cubic regularization. The bounding procedures are general and can be applied to any branch and bound algorithm that uses estimates of the spectrum of the Hessian or derivative tensor, as they are constructed using generalisations of Gershgorin's theorem and other spectral approaches. In order to improve the branching aspects of the algorithm, and using the proposed bounding procedures, we develop parallel variants based on both data parallel and task parallel paradigms and address important performance issues that arise such as doubling of work and load balancing. Numerical testing of the bounding techniques and parallel approaches on a HPC cluster is presented with promising results.