I'd be interested in knowing which password or passwords they are having problems with.
According to an online search, the honolulu.gov domain is managed by the C&C Honolulu. They list 7 name servers and 3 mail exchange servers.
It looks like the Parks & Rec pages were created using the WordPress Content Management System (CMS). The password they "lost" is probably the WP Administrator password.
If they have the database password where the WP content is stored, they could easily log into that and delete the password. That would let them in without typing any password. If they can do that, they can also verify that the administrator login hasn't been changed, which is a standard security step. Giving the administrator login restricted access (like a normal user) and creating a different admin login reduces the risk of someone attacking the default admin login.
Depending on the database engine being used, there are ways to change those passwords as well -- as long as they can log into the O/S.
I've had to reset passwords in many different web sites. It all depends on what you do have access to before knowing where to start.
It might also be as simple as resetting the email password for the former employee's C&C email account, then going to the web interface and starting a password reset. That usually relies on sending an email to the user's email address. There should be email admins there that can access the account, assuming he didn't use a personal account just to f*** with them.
Also, some features in WP save the database login info in clear text, which might be another starting point.