The symptons you describe are quite similar to what happened to me a few years ago. I wouldn't call mine a vibration as much as I would a "rough idle", but could be just semantics.
Try this, assuming you have a TIP. Put the car in gear, put your left foot firmly on the brake so it doesn't run away, note the rpms (should be around 800 give or take). With your right foot on the accelerator, raise the rpm to 1200. Is it any smoother? If so, increase to 1500. Is it smoother still? If so, I'd clean the throttle body really good.
The problem I had was very noticeable in 1st and reverse, but not in the higher gears. This is because once that incredibly filthy throttle body opened up a little (so you could get some air), all was better. It was only at idle or very low rpms where the vibration or roughness was really noticeable. And it was never an issue in Park or Neutral, just like yours.