Many articles have used voice analysis to detect Parkinson's disease (PD), but few have focused on the early stages of the disease and the gender effect. In this article, we have adapted the latest speaker recognition system, called x-vectors, in order to detect PD at an early stage using voice analysis. X-vectors are embeddings extracted from Deep Neural Networks (DNNs), which provide robust speaker representations and improve speaker recognition when large amounts of training data are used. Our goal was to assess whether, in the context of early PD detection, this technique would outperform the more standard classifier MFCC-GMM (Mel-Frequency Cepstral Coefficients-Gaussian Mixture Model) and, if so, under which conditions. We recorded 221 French speakers (recently diagnosed PD subjects and healthy controls) with a high-quality microphone and via the telephone network. Men and women were analyzed separately in order to have more precise models and to assess a possible gender effect. Several experimental and methodological aspects were tested in order to analyze their impacts on classification performance. We assessed the impact of the audio segment durations, data augmentation, type of dataset used for the neural network training, kind of speech tasks, and back-end analyses. X-vectors technique provided better classification performances than MFCC-GMM for the text-independent tasks, and seemed to be particularly suited for the early detection of PD in women (7-15% improvement). This result was observed for both recording types (high-quality microphone and telephone).