How to lay iPhone 13 Pro without case

I'd lay it with the screen up for two reasons:

1. The screen is visible so you can still interact to some extent with the device (get possible notifications)
2. The potentially scratchable area (display + front camera) is way larger if compared to the back side (back camera)

Regardless the side used it's better to avoid rough and abrasive surfaces, of course.

Both sides can be scratched which can compromise the device cosmetically (unpleasant to see) and functionally (uneven and difficult to read screen or scratched camera lenses).

both

if you place it face down it have a different behavior

Quick Tip: Place Your iPhone Face Down to Save Battery Life (howtogeek.com)