If your mom and dad own the domain name, they own the rights to do whatever they want with the web pages. They just need the proper log in information to access their account. The hosting site should be able to provide all that information. If someone has locked them out of their own site and they are paying for it, they need to contact, by phone, the support people at the hosting site and recover control immediately.
You have much better options in using a fully paid hosting site than a free one. You have a complete Control Panel as part of the hosting package.
I hope your mom and dad are not having problems with the designer of your site. Some hold these sites as hostage by refusing to let you know all the account info.
Ron