tent campers tend to not use so much water as an a frame might with ready water access and all the appliances. when i tent camped i tried not to use pots and pans that would need washing if i could help it.
but yes, generally gray water, the mix of water, soaps, grease, food particles etc. are generally not permitted to be run onto the ground anywhere in any campground in the northeast ive been to.
i would not want to be the camper next at the site of anyone who say,washed dishes for 8 for 2 weeks of fatty, greasy foods and left all that stuff on the campsite pad. mushed into the grass, gravel, dirt of whatever the topsoil material was.
if i was the only camper in that spot for the next year, and the soaps were not harmful and grease was not an issue, then i might think about it. but if every camper did it on every spot they camped over an entire season all the sites quality would quickly degrade i think.
i drain into the blue water bucket posted in the earlier post. i dump that at the dump station, as a diswashing station (if equipped and permitted) or into a flush toilet (if equipped and permitted)... its not that big a deal and it keeps the area around the campsite a bit cleaner which isnt a bad thing.