We present a framework to recognize Parkinson's disease (PD) through an English pangram utterance speech collected using a web application from diverse recording settings and environments, including participants' homes. Our dataset includes a global cohort of 1306 participants, including 392 diagnosed with PD. Leveraging the diversity of the dataset, spanning various demographic properties (such as age, sex, and ethnicity), we used deep learning embeddings derived from semi-supervised models such as Wav2Vec 2.0, WavLM, and ImageBind representing the speech dynamics associated with PD. Our novel fusion model for PD classification, which aligns different speech embeddings into a cohesive feature space, demonstrated superior performance over standard concatenation-based fusion models and other baselines (including models built on traditional acoustic features). In a randomized data split configuration, the model achieved an Area Under the Receiver Operating Characteristic Curve (AUROC) of 88.94% and an accuracy of 85.65%. Rigorous statistical analysis confirmed that our model performs equitably across various demographic subgroups in terms of sex, ethnicity, and age, and remains robust regardless of disease duration. Furthermore, our model, when tested on two entirely unseen test datasets collected from clinical settings and from a PD care center, maintained AUROC scores of 82.12% and 78.44%, respectively. This affirms the model's robustness and it's potential to enhance accessibility and health equity in real-world applications.