In Steam, there is a setting that converts controllers to a mouse/keyboard. You should disable this if you are hosting. If this Steam setting is enabled, Steam will always convert your friends controller to a mouse on the desktop and in Non-Steam games.
Go to your settings on Steam.
Go to the Controller Settings
Uncheck this box (note: in Parsec all remote controllers are registered as Xbox controllers, even if they're not one).
It's also possible to disable desktop mode on its own if you want the Xbox support for some reason (controller settings > base configurations > desktop mode > browse configs > disabled).