I'm wondering if there is a way to change a user's password through API?
Apparently, this was possible by using SetUserPreferences (https://www.mediawiki.org/wiki/API:Edit_-_Set_user_preferences), but this is no longer supported. And the API Options feature (https://www.mediawiki.org/wiki/API:Options) doesn't seem to have the option to change passwords.
Is there a way that allows me to do such a thing?
You can do it via the (not very well documented) changeauthenticationdata API. The API is fairly generic as the authentication system might change from wiki to wiki and does not necessarily involve passwords.