You should inform management that as a Scrum Master you have NO 'executive' (for the lack of a better word) power over your team nor are you responsible for your team's performance. You and your team are ONE in front of the management.
The person responsible for the team is the team lead or direct manager (probably a mid-level manager) and only that person can 'get' the team to do something.
As a Scrum Master, you can facilitate and plan these activities with your team. E.g. as a team, you agree that no Story is considered 'Done' unless all story-related activities have been logged into a timesheet. You can be responsible to check that. (My experience is that timesheets are done at the end of the week and are not entered per story but per project).
Or, create an awkward story "As manager I need all my team members to complete timesheets", create subtasks for each member and plan it in the Sprint.
These can be in the scope of your role. You can be responsible to remind your team members to complete their timesheets(e.g. on the deadline day's stand-up meeting).
If a team member keeps failing to enter these as you as a team agreed, you can bring that up in the Retrospective meeting with your team. And, if that doesn't help, escalate to his/her manager.
This is what you can propose to your management as a solution.
The Product Owner, in it's pure scrum role should not be managing the team. It does have full responsibility to manage the product, the sprint and team's effort.