Here's what finally resolved this issue for me:
I VNC'd to my folks' Mac to verify that the shared passwords group did not disappear on their end; Fortunately, it did not. Looking at the "Manage" option of the Family Passwords group on their Mac, I noticed that I had been originally added to the group via my phone number instead of my iCloud email. So, I decided to remove myself from the passwords group via my iPhone, and then had my Dad re-invite me, but using my iCloud email address instead. Here's what happened, after I left the group:
- Any passwords that I had shared were unshared & moved back into my personal passwords (fortunately, out of the 100-some passwords, I had only contributed ~20)
- The Family Passwords shared group disappeared from my iPhone/iPad (it was already MIA on my Mac)
- I then had my Dad re-invite me to the group, using the primary email address associated with my Apple account
- As soon as he sent the invite, a "Pending Invites" section appeared on all of my devices within the Passwords app (including my Mac!)
- Using my Mac, I accepted the invite, and after a few seconds, the Family Passwords group re-appeared on my Mac, as well as on all of my other devices
- I then re-added the passwords/accounts that I had originally shared with the group; They synced to all of our devices almost instantly
It's been ~24 hours & the shared passwords group is still present on my Mac (I've rebooted several times as well, just to be sure that doesn't cause it to go away). So, in my case, I'm not exactly sure whether simply leaving & being re-added to the group resolved this; Or perhaps being re-added, but using my email instead of phone number resolved it. Just figured I'd loop back here to post my findings, in case this helps anyone else!