My Controller Is Recognized As Mouse/Keyboard On Host

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.

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Go to the Controller Settings

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Uncheck this box (note: in Parsec all remote controllers are registered as Xbox controllers, even if they're not one).

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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).