Strange sound coming from under car? 2001 Mustang GT

Hard to say from the video, but I'd start by being suspicious of an exhaust leak, then the throw out bearing if a stick. Get a piece of garden hose or 3/4" heater hose as long as your arm plus four or so inches...hold one end to your ear and slowly move the other around all the exhaust pipe connections, transmission, and also do under the hood...sound radiates---the hose will conduct the sound to your ear helping you locate it. You can do a similar check with a long screwdriver, breaker bar, etc when looking for mechanical noises---touch one end to your ear or temple and the other to the mechanical item you wish to check---I do this for rocker arms, water pump, alternator, ac comp, ps pump, etc.

Other thoughts...since the car is stationary, it pretty much rules out anything aft of the spinning pieces in the transmission except exhaust...the input shaft on a manual shift, and the torque converter/input on an auto spin while in neutral, so give that area a listen. If the shield is off the lower front of the trans bellhousing, you may have picked up a piece of trash that is making the sound (remote, but possible)...you should be able to hear it with the hose. Additionally, if a manual shift, your throwout bearing may be getting noisy...make sure brakes are set and have someone push in and let out on the clutch pedal to see if it changes the sound (do not put it in gear while you are under it).

Oh, and listen to the fuel pump as well. Happy hunting!