I have an 2006 Audi A4 Quattro with 72,000 miles. I just bought it a month ago.

There is a loud squeaking noise coming from the engine. At first I thought it was the belt, but belt dressing did nothing to fix it. It only screeches sometimes.

It screeches when:
I am driving at cruising speed.
When the car is in park.

It stops screeching when:
I accelerate when driving. If I let up on the gas it screeches again.
When I come to a stop light and idle it stops screeching.
If I park and keep the car in drive and hold the brake the sound stops, but if I turn off the A/C while doing this the sound starts up.

When I turn the engine off it squeaks for a second or two then stops. I have watched the main belt durning shutdown and it stops turning, but the sound continues for a second and ends.

I have posted a video of the sound while parked and when I shut the engine off. Here is the link. Audi A4 - YouTube

Does anyone have a diagnosis?