I've never seen it spelled this way. There are numerous references to padiddle in various sources. We had a slightly different version in high school. The last person to say "padiddle" had to remove an article of clothing.
I used to like this word--to me it means a car with one headlight out--until, every time I said it when I saw said car, my S.O. would reply (in like volume) "Pimply butt!" I failed to see the humor, tried explaining, tried getting mad about it, and finally just stopped giving a shit. Now I just don't say it anymore.
We played a lot of car games. My favorite to play was one my dad made up. There were a lot of curvy two-lane roads to drive on, so when it was dark, we'd have to try to say "dim!" before my he could flick the dimmer switch on the headlights. You lost points/cred if you dimmed stationary lights or your own reflection on the river. And he wonders why I'm such a micromanager.