Learn English – Grammaticality: ‘neither “but” nor “yet” make apt conjunction’

You aren't talking about the parties together as a unified group, but how they feel individually. You generally are always doing this when you use neither.

This may help you understand:

Neither [one] of the parties is concerned with Egypt

Usually the one is understood and left out of the construct.

I'm not sure if the below is correct, but here goes:

Let's say you had a total of four parties, and for some reason, they were afflilated in two groups, i.e. party A and B, and party C and D. Then you could say:

Neither two of the parties are concerned with Egypt.

You can see how this is confusing unless there's a strong context to help, so I would really avoid using neither in this situation, even if it is correct.

A comma after a coordinating conjunction (and, or, but) is only acceptable when the comma is the first of a pair of commas bracketing a ‘supplement’—a phrase or clause which lies outside the main structure of the sentence and interrupts the flow of thought.

  And, as if that were not enough, he went on to sneer at the victim.
     ^---------------------------^  

  But, as you said, there is no point in trying too hard.
     ^------------^

In other circumstances the comma is effectively ‘disjunctive’ and subverts the joining effect of the conjunction.

This counterproductive comma is becoming very common in contemporary business writing; I suspect it represents a hypercorrection by people who have misunderstood the need for commas bracketing supplements, or think it lends formal ‘elegance’ to their writing.