Is running AES NI in parallel possible (one encryption per core/thread)?
It depends mostly on the memory bandwidth. Each core contains the dedicated AES hardware, but the CPUs need to obtain the data to encrypt or decrypt somehow. Even with a single core you have to be careful about how you dispatch instructions to get maximal performance, see Intel's white paper, page 48 bottom.

If you already have the machine, you can always try it out and let us know how it went.