Can AES ans SHA-2 algorithms be implemented in parallel? As I understood AES GCM mode of operation gets good performance result because parallel execution of both encryption and hashing part. 

Why AES and SHA-2 computation cannot be pipelined ?
They can be computed in parallel.

But what would still make AES-GCM faster than AES-SHA2 is GCM itself.

GCM is a universal hash function: it can be very fast, especially on CPUs supporting carry-less multiplications.
But it doesn't satisfy the same properties as a hash function such as SHA2. In particular, not encrypting its output would have catastrophic implications. It's fast because it is a simple and specialized function. 

Even with Intel and ARM SHA extensions, I doubt it would be faster than GCM.

With AESNI and CLMUL, AES-GCM is about 0.65 cpb. SHA256 alone using Intel extension is about 3.8 cpb.