For the hash computation of SHA-256, is it computed in parallel or series?
SHA-256 has a basic building block which is not easily paralalized. Between blocks SHA-256 follows a Merkle-Damgard construction and needs to be computed sequentially.
You could build a tree-hash using the same compression function, and this will be secure and allow parallel computing but results would not be consistent with the SHA-256 standard. 

https://en.wikipedia.org/wiki/Merkle%E2%80%93Damg%C3%A5rd_construction
https://en.wikipedia.org/wiki/Merkle_tree