Although not techincally within Zion National Park, this popular canyon will require a Zion National Park permit when combined with Orderville and the Zion Narrows - which is the typical way to do this canyon. Hence the parent for this canyon page is the Zion National Park page.
"Imagine an impossibly narrow canyon with sinewy curved and twisted wind-swept paths, lit with mysterious shadows and hints of light, that tempt canyoneers to wiggle, slither and climb to its finish. Birch Hollow is such a canyon, yet it's a forgotten gem in the canyoneering world, because it doesn't hold the stigma of being within the Zion National Park boundary. Not only does Birch Hollow offer some of the best canyoneering to be had anywhere, it can be combined with two of Zion's magnificent classic canyons, Orderville Canyon and the Zion Narrows", see Zion's Birch Hollow
I hiked/rappelled Birch Hallow Canyon ending up in Orderville Canyon, and finally into the Virgin River Narrows. Birch Hallow involved 10 rappels with one as long as 100 feet. There were at least 2 rappels in Orderville Canyon and several tricky spots. The Narrows were a beautiful hike to the Zions Shuttle at The Temple of Sinawava.
"Leave the park [Zion] from the east gate and drive 2.4 miles. Turn left at the junction of North Fork Road and SR-9. Drive 5.2 miles up the paved road to the Ponderosa Ranch. Don't drive in, but re-set your odometer and continue on the now dirt North Fork Road (may be impassible when wet). When the odometer hits 3.2 miles you are at Birch Hollow. There is a place to pull over on the left side of the road. The wash on the left is where the hike begins", see Zion's Birch Hollow.
The trailhead, point A (see image below) is at: N 37.30973 W 112.85637.
On the image above Birch Hollow begins at point A - about 8 miles up the North Fork Road on the east side of Zion National Park. The canyon heads NW for about 2 miles ending at point C. It involves 10 rappels, with the longest one being 100 feet. It connects with OrderVille Canyon at point C, which if followed east will enter into Zion National Park. If followed west, it can be used to return to North Fork Road. It's about 3 miles from North Fork Road back to the Birch Hollow trailhead. After about 5 or 6 miles Orderville enters the Narrows, at point B. Follow the Virgin River, within the Narrows, downstream (southernly) to the Temple of Sinawava 2 to 3 miles distant.
An Orderville Canyon permit is required to exit down the Orderville Canyon to the Zion Narrows. No permit is required if exiting up the Orderville Canyon.
Good luck getting a camp spot inside Zion NP.