Additional features, settings, or issues not covered by specific Microsoft Teams categories
It's configurable. By default, organizers, co-organizers and anyone with the presenter role can start the recording/transcription. If you want to amend that, use the Meeting options dialog > Recording & Transcription > Who can record and transcribe:
In your case, make sure the person you want to start the recording has one of the corresponding roles assigned. There are also some admin controls, but the above should be sufficient for most needs.