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Parkinson's Disease (PD) is a neurological disorder that affects facial movements and non-verbal communication. Patients with PD present a reduction in facial movements called hypomimia which is evaluated in item 3.2 of the MDS-UPDRS-III scale. In this work, we propose to use facial expression analysis from face images based on affective domains to improve PD detection. We propose different domain adaptation techniques to exploit the latest advances in face recognition and Face Action Unit (FAU) detection. The principal contributions of this work are: (1) a novel framework to exploit deep face architectures to model hypomimia in PD patients; (2) we experimentally compare PD detection based on single images vs. image sequences while the patients are evoked various face expressions; (3) we explore different domain adaptation techniques to exploit existing models initially trained either for Face Recognition or to detect FAUs for the automatic discrimination between PD patients and healthy subjects; and (4) a new approach to use triplet-loss learning to improve hypomimia modeling and PD detection. The results on real face images from PD patients show that we are able to properly model evoked emotions using image sequences (neutral, onset-transition, apex, offset-transition, and neutral) with accuracy improvements up to 5.5% (from 72.9% to 78.4%) with respect to single-image PD detection. We also show that our proposed affective-domain adaptation provides improvements in PD detection up to 8.9% (from 78.4% to 87.3% detection accuracy).