I went to a talk given by the man who developed Parseltongue for the Harry Potter films, Prof Francis Nolan. Just a few ‘facts’ about the language with some of the ‘explanations’ given:
Phonology
It’s got no rounded vowels or labial consonants (because snake lips aren’t very flexible)
It’s got pharyngeal consonants (because some snakes like to constrict things)
It’s got a large number of fricatives, which also exhibit a length contrast (because…snakes)
Syntax
It’s got basic VSO order
It’s got postpositions (typologically highly unusual for a VSO language)
It’s ergative
Borrowings
The word ‘muggle’ has been borrowed into English from Parseltongue ‘ŋaʔalas’ – obviously!
For more information, check out this guide to learning Parseltongue, this grammar, this Parseltongue-inspired grammar, or this translator, although I’m not sure how authoritative they are.