Answers to Some Logistical Questions

Where are Buckskin Gulch and Paria Canyon? In far southern Utah and northern Arizona. The closest town to the start is Kanab, UT.
What route did you take? There is only one route once you are in the gulch itself. We came in from Wire Pass and hiked down Buckskin Gulch to Paria Canyon. We swung south from there and hiked down to the Adams Cattle Trail. We took this back up to the plateau and then hiked cross country across the Sand Hills back to Wire Pass via Coyote Buttes. The map below (from Google Earth) shows our approximate route.
How do I get to Wire Pass? Drive US89 east from Kanab, UT for about 50ish miles. There is a sign for House Rock Valley road on the right. Take that. The road isn't paved, but my Honda Civic cruised it without issue. I wouldn't want to try it after a good rain, though. Wire Pass is a big trail head complete with outhouses (no water) about 9 miles from US 89.
Who manages the land? Do we need permits? The land is within, mostly Grand Staircase - Escalante National Monument and is managed by the BLM. Most of the area is designated wilderness. You need permits for Buckskin Gulch and Paria Canyon. There is a quota for how many people can camp overnight in the area. You can apply and get permits online, but I don't remember the link at this moment. As of this writing, permits were $5 per person per day.
How far did you hike? Not far. The first day was 15 miles from Wire Pass to the confluence with the Paria. The second day was about 8 miles. The third was about 15 or more. The third maybe 10. Except for the first these are wild estimates.
What about water? Buckskin has some nasty cess pools that you might be able to get water out of with a filter, but it is better to bring your own. Near the confluence you can find small streams of sweet water. The Paria is a muddy mess, so be prepared to spend a lot of time filtering, or letting the water settle overnight in a jug. There are some springs, which are pretty obvious. Big Spring is the closest one that we saw to the cattle trail and it is several miles away. We found no permanent water sources on the plateau, but there are cows up there and they must drink something.
Any special dangers? Quicksand can be a real problem as we found out. I wouldn't want to be a solo hiker in there. Only go into Buckskin if you have a solid weather forecast. If a flash flood comes through there is nowhere to escape to and you will die.
Is there anywhere to camp near Kanab? We asked lots of locals and no one knew anything. We were referred to the RV park in town. Instead, about four miles north of town on US 89 there is a sign and a paved road for Kanab Canyon. Follow it down and turn off to the right on a good dirt road. Drive back a few yards and you'll have an excellent campsite.
Is there anywhere to camp near Wire Pass? Just camp at Wire Pass. If you want a picnic table and outhouses, instead camp at Whitehouse, which is located about 5 miles further east of US 89 than the turn off for Wire Pass. There is a ranger station there that held very uninformed and unhelpful rangers and volunteers, but there is water. It tastes salty, so bring your own. The road to the campground is not paved and a sign was up saying that it was impassable due to a washout. There was no washout and my Civic made it just fine. It costs $5 to camp there. You can also access Pariah Canyon from here.
What is this whole carrot thing? I'm not telling and neither are the others.