Analyzing the probabilities of ordinal patterns is a recent approach to quantifying the complexity of time series and detecting structural changes in the underlying dynamics. The present paper investigates statistical properties of estimators of ordinal pattern probabilities in discrete-time Gaussian processes with stationary increments. It shows that better estimators than the sample frequencies are available and establishes sufficient conditions under which these estimators are consistent and asymptotically normal. The results are applied to derive properties of the Zero Crossing estimator for the Hurst parameter in fractional Brownian motion. In a simulation study, the performance of the Zero Crossing estimator is compared to that of a similar "metric" estimator; furthermore, the Zero Crossing estimator is applied to the analysis of Nile River data.