Depending on whether the game is being hosted in the Computers tab or Arcade, the resolution is changed in different ways. In both modes, you can check the current resolution by looking at the Parsec overlay in the guest.
Desktop mode (Computers tab)
The resolution of the stream is always the same as resolution of the host's display that is being captured. Parsec does not downscale the resolution for the guest, and any settings in Parsec will only adjust the host display's actual resolution.
When you're the owner of the computer, you change the resolution from the Parsec overlay, or set the resolution the primary display will use when you join, in the Parsec host settings. If your hosting computer does not support the resolution that you're requesting based on the Windows Display Settings, Parsec will switch to the closest resolution available. To add a custom resolution to the host computer, use the Windows Display Settings or the AMD/NVIDIA control panel.
If a friend is joining, the host settings in Parsec will not take effect and they won't be able to adjust the resolution from the overlay. You will want to change it on the host's computer directly in the Windows Display Settings.
Arcade captures just the game, so the resolution will be the size of the game window. If the game is in fullscreen, it will be the resolution of the display. Usually you'll want to change the resolution through the in-game settings, or by manually resizing the window by dragging the sides of the window.
Keep in mind that some games do not change the resolution of the display itself in fullscreen. It's possible that, for example, the game is rendering at 1280x720, but is displaying that at 1920x1080, which is the resolution Parsec will use. When in doubt, ask your guests what resolution they see in their Parsec overlay. You can change your display's resolution in the Windows Display Settings in case your game is not changing it for you.