Pure soil wouldn't be a good option. Anything that's subject to high traffic, whether foot or vehicular, will also be subject to soil compaction which implies water pooling. That's a long way of saying those parking lots will turn into mud lots. At a minimum you could grade and gravel them, going the extra mile to crush the gravel so it stays in place longer. I do this with the driveway at my house because it rains a lot where I live. Paving would arguably last a lot longer and probably be more bang for my buck, but not having a place for water to go and having a basement is not a good combination.

On my driveway we only tend to get clovers and really resilient weeds. So, either really shallow rooted plants or weeds with absurdly strong roots. You're not likely to get native plans to grow unless you section off parts of the area for maintaining living soil.

Unpaved parking lots tend to lose their plant growth when frequently used, after which the parking lot turns to mud in wet weather, which then results in stuck cars and dirt getting into stores with the customers that walked through that mud.

To give a visceral example- There's an episode of Clarkson's Farm where the customers that parked in the grass at his farm shop need to get their cars pulled out of the mud by a tractor.