Do I need to get a parking permit? Which area is closest to the falls?

You absolutely need a permit.

Not true, you DO need a permit to park anywhere in the fossil creek area, there will be forest service personel that will stop you and check for your permit. They can be purchased online for $6 from the recreation.gov website.The waterfall trailhead lot is the closest to get to the main waterfall.

You do not need a parking permit. The closest to the falls is the parking lot that's furthest down the road. It makes sense when you get there! Also, the road is very Rocky and not paved.