Parkinson's Disease (PD) is a neurodegenerative disorder characterized by motor symptoms, including altered voice production in the early stages. Early diagnosis is crucial not only to improve PD patients' quality of life but also to enhance the efficacy of potential disease-modifying therapies during early neurodegeneration, a window often missed by current diagnostic tools. In this paper, we propose a more generalizable approach to PD recognition through domain adaptation and self-supervised learning. We demonstrate the generalization capabilities of the proposed approach across diverse datasets in different languages. Our approach leverages HuBERT, a large deep neural network originally trained for speech recognition and further trains it on unlabeled speech data from a population that is similar to the target group, i.e., the elderly, in a self-supervised manner. The model is then fine-tuned and adapted for use across different datasets in multiple languages, including English, Italian, and Spanish. Evaluations on four publicly available PD datasets demonstrate the model's efficacy, achieving an average specificity of 92.1% and an average sensitivity of 91.2%. This method offers objective and consistent evaluations across large populations, addressing the variability inherent in human assessments and providing a non-invasive, cost-effective and accessible diagnostic option.