Your access to project timesheets depends on whether you're:
An admin
Project manager
Employee
Admin
An admin can create projects, assign employees, and do anything and everything when it comes to processing project timesheets.
Project manager
When you assign someone as a project manager of a project, they can create and edit work packages within the projects they manage. They can also submit, approve, and decline project timesheets.
Employee
Employees can log project hours against the projects under the work packages to which they're assigned. They can submit these for approval by the project manager.
📎NOTE: There are no permissions that you can configure for roles regarding timesheets or project timesheets.