Rearranging the controller order can be useful in various scenarios, such as when your game only lets one person control at a time and you want to take turns with someone else. While Parsec itself does not have an easy feature that lets you change the controller position, you can still do it with either method below. You should do these on the computer that is hosting the game.
Method 1: Use the game or emulator settings
Most emulators have flexible enough settings that allow you to change who is in control of which slot in the emulated console. Some games may also let you do it, although that is rarer. Check your game or emulator settings.
Method 2: Disconnect and reconnect in order
You can turn off all guest and host controllers, then reconnect them in the order you want them to be in, such that the game recognizes controllers in a different order.
The host should unplug or turn off their own controller, and right-click Parsec in the taskbar and click "Reset gamepad" to disconnect the controllers of all the guests who are connected. If the game is being hosted on Arcade, the "Reset gamepad" function will be in the in-game overlay.
To reconnect, plug or turn on the host controller, and press buttons on the guest controllers in the order you want the controllers to be in. Once done, you should have a different controller order.
Method 3: Steam Input
You can start a game inside Steam's Big Picture Mode, and use the in-game big picture overlay to change the position, or configure a button combo on your controller to do it.
After starting a game inside Big Picture, you can use the guide button (xbox or playstation icon in your controller) to bring up the overlay, and in Controller Options > Rearrange Controller Order, you can tweak the order.
If you want to set a button combo in your controller to more quickly switch the position, you can make it so that pressing the guide button along with something else swaps your controller into a given position.
To configure it, get out of the game and go back to the main Big Picture menu. Click the Settings icon > Base Configurations > Steam Button Chord. Select something like button X, click the icon that has a controller and two users, and select the player that your controller will be swapped to when you press the guide button along with the button you chose. Once done, open a game inside Big Picture to use it.
You'll probably want at least two different buttons, for example, Guide button + X set to swap your controller to player 1, and Guide button + B set to swap your controller to player 2.