Is there any particular reason that you want to self-host? Bitwarden’s staff and subcontractors have the expertise to ensure the servers are secure from attacks, and that any server downtime is minimized; the conventional wisdom is that “if you have to ask, you’re probably getting in over your head”.
To answer (one of) your questions: yes, you can use your own custom domain name. I believe that the Bitwarden server needs to be set up on a subdomain of your domain, though.
If you have shared passwords (or other shared vault items) that all three of you will need access to, then a Family plan subscription is needed. If you have your own passwords, but your parents want to share passwords with each other (but not with you), then the two of them can become members of a free organization (for sharing) after signing up for individual accounts (whether free or Premium).
My parents and passwords are a no go @grb. I got the family plan so that I can manage their passwords and acces them if needed. They can acces their own passwords and the ones both need, but if they need to do it themself it will be everywhere: qwerty, 123456 or stuff like that.