The demo only has support for 1 controller at a time so if you are wanting to use multiple controllers, whether locally or remotely, you will have to buy the full version.
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So the way you can control your partners gamepad is by clicking the onscreen controls with your mouse; that will control your partners gamepad . With the demo, the onscreen controls are locked so you will need the full version. If Parsec is not working for you either after this, you can always try Steam Remote Play too. Buying from Itch will also give you a Steam key so you can use either one.