Parking brake and esp light definitely can be linked to it. I'd recommend getting the battery tested and replace if found defective bas a starting point.
Sorry to hear of your issue. If the garage haven't already, it might be worth getting a diagnostic check done to see if any fault codes flag up. Also you could consider a battery test as a failing battery can prevent the stop/start from.working.
We're doing black forest, the ring, all the German manufacturer museums. Taking the A6 back to where it was built 20 years ago. Then going to colditz and some other WW2 sites. Then getting ferry back from hook of Holland.
If you have the function it'll either be in the mmi settings or on a switch on the back of each headlight. I'm going to Germany in September. Where are you going in Europe?.
If you bought it less than 30 days ago then you can get the dealer to fix it. You could get a diagnostic check carried out to see what fault codes flag up. It could be a failed compressor so expect an expensive bill if so.