macOS Catalina 10.15 introduced more privacy restrictions, without your permission applications simply cannot access every folder. Therefore Parallels Access needs additional access to be able to work as before.
List of required permissions for Parallels Access macOS agent
Accessibility: Parallels Access requires this permission to use Accessibility features to control computer’s applications by using the app on mobile device.
Screen Recording: Parallels Access requires this permission capture content on computer’s screen and transmit it to user’s mobile device.
Full Disk Access: Parallels Access requires this permission in order to provide functionality in File Manager: to browse, view and open on a mobile device files located on remote computer.
Additional software for sound: Parallels Access installs additional software to provide ability to receive sound from remote computer on mobile device, and send sound from mobile device's microphone to computer when in remote session - when sound from mobile device needs to be passed to a remote computer.
List of required permissions for Parallels Access iOS client
Bluetooth: Parallels Access iOS client requires this permission to use Swiftpoint GT mouse.