Is running AES NI in parallel possible (one encryption per core/thread)? AES NI seems to perform AES operations much faster than doing in software. However, if I have a machine with a large number of cores (say 32 cores), can I perform 32 AES encryptions using AES NI instruction i.e. one per core? Or can I just run one AES instruction?
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.