Human-Computer Interaction (HCI) and games set a new domain in understanding people's motivations in gaming, behavioral implications of game play, game adaptation to player preferences and needs for increased engaging experiences in the context of HCI serious games (HCI-SGs). When the latter relate with people's health status, they can become a part of their daily life as assistive health status monitoring/enhancement systems. Co-designing HCI-SGs can be seen as a combination of art and science that involves a meticulous collaborative process. The design elements in assistive HCI-SGs for Parkinson's Disease (PD) patients, in particular, are explored in the present work. Within this context, the Game-Based Learning (GBL) design framework is adopted here and its main game-design parameters are explored for the Exergames, Dietarygames, Emotional games, Handwriting games, and Voice games design, drawn from the PD-related i-PROGNOSIS Personalized Game Suite (PGS) (www.i-prognosis.eu) holistic approach. Two main data sources were involved in the study. In particular, the first one includes qualitative data from semi-structured interviews, involving 10 PD patients and four clinicians in the co-creation process of the game design, whereas the second one relates with data from an online questionnaire addressed by 104 participants spanning the whole related spectrum, i.e., PD patients, physicians, software/game developers. Linear regression analysis was employed to identify an adapted GBL framework with the most significant game-design parameters, which efficiently predict the transferability of the PGS beneficial effect to real-life, addressing functional PD symptoms. The findings of this work can assist HCI-SG designers for designing PD-related HCI-SGs, as the most significant game-design factors were identified, in terms of adding value to the role of HCI-SGs in increasing PD patients' quality of life, optimizing the interaction with personalized HCI-SGs and, hence, fostering a collaborative human-computer symbiosis.