Within the Bayesian framework, we utilize Gaussian processes for parametric studies of long running computer codes. Since the simulations are expensive, it is necessary to exploit the computational budget in the best possible manner. Employing the sum over variances —being indicators for the quality of the fit—as the utility function, we establish an optimized and automated sequential parameter selection procedure. However, it is also often desirable to utilize the parallel running capabilities of present computer technology and abandon the sequential parameter selection for a faster overall turn-around time (wall-clock time). This paper proposes to achieve this by marginalizing over the expected outcomes at optimized test points in order to set up a pool of starting values for batch execution. For a one-dimensional test case, the numerical results are validated with the analytical solution. Eventually, a systematic convergence study demonstrates the advantage of the optimized approach over randomly chosen parameter settings.