@Mike_Pat Welcome to the forum!
Technically, Bitwarden’s Terms of Service specify that “a single login may not be shared by multiple people”, so to be compliant with the terms, your parents would each have to have separate individual accounts (which are then each joined to your Family plan), but other than that you should be able to do most of what you described in your post.
As a organization owner/admin, you will always have a way of accessing vault data stored in any of the organization’s collections. A collection’s access permissions could be configured so that you do not see any of the collection’s contents while you are using your own Bitwarden apps — however, you would still be able to go into the Admin Console (accessible from the Web Vault only), where you will have the ability to see collections for which your user account has not been given view access.
You can certainly set up permissions to prevent your siblings and parents from viewing any collections that are shared between you and your spouse, and since only you are the organization admin, they will not have access to the Admin Console “back door” for accessing these private collections.
The Family organizations do not offer any method for admins to “recover” other members’ individual accounts (e.g., if they lose access to their master password or 2FA), so you don’t have to worry about that — the individual vaults will always be safe (as long as each user has a master password that is unique, confidential, and randomly generated, has enabled 2FA, has good security for their devices, and practices good internet hygiene).
I hope this answers your questions!