My 340 does the same thing. I traced it back to the water pump and fan through process of elimination. Mine would do it weather it was in park or going down the road.
I replaced the water pump and it got better. I took off the flex fan and put on a clutch fan. When I did that the pulsating vibe changed to a less more speratic vibe, but much less noticable. If I take off the fan completely the vibration goes away. So a quick easy test is take off the fan and put bolts in to hold the pulley on. This is if it vibrates in park .
I replaced the water pump and it got better. I took off the flex fan and put on a clutch fan. When I did that the pulsating vibe changed to a less more speratic vibe, but much less noticable. If I take off the fan completely the vibration goes away. So a quick easy test is take off the fan and put bolts in to hold the pulley on. This is if it vibrates in park .