First, search the help docs and this forum. Maybe your question has been answered! The debugging steps can help, too. Still stuck? Delete this line and proceed.
What I’m trying to do
The tasks in the “Family” task list can be edited/completed by all family members with granted access.
There is probably not a way you will like. Any solution I can think of offhand will give them access to at least the whole file and possibly the whole vault.
The closest thing I know of is if your vault is in Dropbox you can use Dropbox to share the file with them — but they’ll be able to see and change the whole file, not just the task, and they’ll need to open the file in a Markdown editor that recognizes checkbox lists.
With Obsidian Sync you can share a vault with others, but it’s the whole vault and they each need their own Sync subscription. Obsidian Publish has a collaboration feature that doesn’t require everyone to have their own subscription; I don’t remember the details.