Unable to Edit a Coworker's Tasks and Events in Salesforce

Editing Based on Role Hierarchy

Someone who is higher in the role hierarchy can edit tasks and events assigned to the users who report to them in the hierarchy, but subordinates cannot edit activities assigned to someone higher in the hierarchy or activities assigned to users with the same role.

Editing Based on Sharing Settings

If your organization has set Activities to Controlled by Parent in your sharing settings, then users can edit Tasks and Events owned by other users provided they are able to edit the parent records associated with the Activity.

Editing Your Own Assigned Tasks

All users can edit, close, and delete tasks that have been directly assigned to them, regardless of role hierarchy or sharing settings