Underground parking garages are very common over here. Most of the times these city squares are exactly that, a huge multi-level underground parking garage because these squares are always event spaces, and they are usually city-center so even when there isnt events, people have somewhere to park when just visiting the city. Yes, there will even be long lines of traffic waiting/hoping for a spot during event periods.

With that said, they do fill up, usually fast. So most events suggest finding public transportation. This just means people park further away and then take the bus/rail/etc the rest of the way. These Markets arent just for the locals, people travel from all over to come to them. So public transportation for long-distant travel, while totally possible, isnt always as practical (sometimes nor affordable or possible) for everyone. Plus, long distance trains do sell out. We just spent most of the season traveling all over Central Europe going to various markets.