2000+ Audi's don't have an ignition on feed to the HU. They use the CANBUS to tell the HU when the ignition is on, and when the keys are pulled out.
You need to either pair the permanent live with the ignition on feed to the new HU and turn it on/off manually, or create an ignition on feed for the new HU.
Unfortunately there is no easy feed to tap into. They are all either perm live or only live when ignition is fully on i.e. they go off when you turn the ignition off and don't stay live until you pull the keys out, which is what you really need.
If you don't mind the HU turning off when you turn the engine off then you can tap a feed at the fusebox, there are plenty of options there.
If you want a proper ignition on feed i.e. the HU stays on until you pull the keys out, then you need to wire in a relay. The trigger to use for the relay is under the steering column. Remove the trim under the steering wheel and you will see the main loom from the ignition switch running under the left side of the steering column. You'll see some black trunking protecting the loom. Release this trunking so you can get to the wires. You'll see a very thin red wire in amongst all the other thicker wires. This is the one you want.
With the ignition off and the keys out, chop this wire and reconnect it with a small terminal block along with another wire that you can then run to the relay. Unfortunately the wire is so thin that you can't get a reliable feed using a piggyback connector.
This is a very low current feed so you can't power the HU directly from it. Just wire in a 12v relay using piggyback connections from the perm live & earth at the HU and use this new wire to switch the relay on/off, then use the other side of the relay to power the HU. Sounds more complicated than it actually is, took me 30-45 minutes to do mine, but well worthwhile otherwise you need to have the keys turned to ignition on to get the HU to work when parked up.