[Bruan]

Does anyone know how parallelism is achieved in highly serial ILU and still gives relatively good performance?

[pcg]

In SU2 (and many codes) ILU (and LU-SGS) operate only on the local matrix of each MPI rank.
This is called "Additive Schwarz" (makes it sound less hacky if you write a paper).
That is, it is not the ILU of the entire global problem. But linear systems are still solved for the global problem that is why we get away with this approach.
However, for the same tolerance the systems take a few more iterations to solve once you have many cores.
The higher the CFL the worse this gets, and for elasticity problems it becomes very noticeable.
There are ways to compute and apply the full ILU in parallel by transforming it in an iterative problem (a kind of Jacobi iteration to compute it, and then to form the solves with L and U parts).
These approaches are common for massive parallel architectures (GPU's) but they would not perform well on CPU due to much less memory bandwidth (unless you use a number of CPU comparable to the number of threads in a GPU).