I agree the website does not say parking is fee with an HA tag/plate. However, there was at least 1 poster back in the spring or early summer who received an email reply from WDW which stated self-parking is free with an HA tag/plate. I don’t believe anyone has given first-hand account after returning from a visit.
My advice — plan on paying, and maybe you’ll find it isn’t charged.
Enjoy your vacation!