I see the same things. The only permission Threads asked for was Notifications.
I think the poster is confused between technical system-granted permissions (e.g., Contacts, Location, Bluetooth, etc.) and app privacy labels (i.e., a self-declared user-friendly version of a legalese privacy policy).
When I go to Settings > Privacy & Security > Health, Threads is not listed as an app that has access.
Maybe at some point in the future it will request access, and you can just deny it then if you don't want it.