According to Fundéu (Foundation for Urgent Spanish), the surname Parkinson is written with a capital initial if it is preceded by disease (Parkinson's disease) and with a lower case when it is alone, used as a common name (suffers from Parkinson's).
According to the RAE Pan-Hispanic Dictionary of Doubts, the expression "Parkinson's disease" must respect the spelling of the surname of the English doctor who first described this neurodegenerative disease, and write it with an initial capital letter and without an accent mark. In contrast, if the term is used in isolation to refer to the disease, it becomes a common noun ( Parkinson's ), as in 'Parkinson's is a neurodegenerative disorder'.
As for the expression World Parkinson's Day (which is celebrated every April 11), it is written with capital initials, a rule that applies to all nouns and adjectives that are part of the name of festivities, whether civil, military or religious. And with a tilde, for referring in this case to the disease and not to its discoverer.
