We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned Hankel matrices. It is based on the LDLT decomposition and involves finding a k× k sub-matrix of the inverse of the original N× N Hankel matrix H N− 1. The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitutes a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w (x)= e− x β, supported on [0,∞) and β> 0. Such weight generates a Hankel determinant, a fundamental object in random matrix theory. In the situation where β> 1/2, the smallest eigenvalue tends to 0 exponentially fast. If β< 1/2, which is the situation where the classical moment problem is indeterminate, then the smallest eigenvalue is bounded from below by a positive number. If β= 1/2, it is conjectured that the smallest eigenvalue tends to 0 algebraically, with a precise exponent. The algorithm run on the HPCC producing a fantastic match between the theoretical value of 2/π and the numerical result.