In the context of computational fluid dynamics (CFD), a standard method that can assess the accuracy of the solution remains a challenge. In our previous work, we have explored an optimized extrapolation method to improve Richardson extrapolation for solution verification [4, 5, 6, 7]. In this paper, we extends the method to the evaluation of the conditioning number of the PDE at hand. Our method allows the construction not to require an internal knowledge of the software neither the source code that produces the solution to be verified. One of the key aspect of our method is its intrinsic parallelism. We will present in detail an implementation that can take advantage of this parallelism on an heterogeneous network of computers.