Adding a decent layer of clean gravel every year after evening out the new "sub-layer" will help: the gravel will move around a lot and slowly disappear. Wherever soil's exposed, stuff'll grow. Organic matter and seeds will blow in whtever you do, so I think you'll find it a real challenge managing it organically to 'conventional' aesthetic standards, and one reason why glyphosate's so popular!
Of course, potential vehicular stickage! Maybe check the maximum recommended grave depth in a carpark and ensure that's maintained. All theory though... Industrial-strength vinegar is relatively effective. It doesn't kill roots though, so you'd need to pull any biguns at the start, then hopefully spraying will keep on top of the littlies.
In a situation like that, the propane burner is probably your best option. Just make certain that you fry each and every one before it goes to seed. (Most weed seeds can survive a forest fire!) You cannot control seeds brought in by the winds and bird poop, but if you can control the existing weeds before they reseed, it can be a manageable task. Every time you need a break, just tell the boss "I gotta go check the parking lot...I thought I saw some birds setting up camp there!"
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100 ways to cut one's personal carbon footprint - in order of tons of carbon