Simple candidates for pseudorandom permutations?
Yes.  The following paper presents a candidate for a PRF that is implementable in $NC^1$, whose security is based on a lattice assumption (hardness of LWE):

Abhishek Banerjee, Chris Peikert, Alon Rosen.  Pseudorandom Functions and Lattices.  EUROCRYPT 2012.

It also has some discussion of related literature that might be helpful.



Also, here are two trivial observations.

First, there is a PRP that can be computed in $NC^1$ if and only if there is a PRF that can be computed in $NC^1$.  The "only if" part is immediate, as any PRP with large domain is also a PRF.  The "if" part follows from the Luby-Rackoff construction (i.e., the Feistel cipher), as that shows how to build a PRP out of any PRF; it increases the depth by only a constant factor.

Second, the following paper shows that no PRF can be computed by an AC0 circuit.

Nathan Linial, Yishay Mansour, Noam Nisan.  Constant depth circuits, Fourier transform, and learnability.  Journal of the ACM, 40(3):607--620, 1993.

It follows that no PRP can be computed by an $AC^0$ circuit.