punctuation - Comma after a coordinating conjunction preceding a parenthetical at the start of the sentence

Although similar questions have been asked before, I am still not clear as to official or, at the very least, preferred position from punctuation rules point of view on comma after coordinating conjunction that precedes a parenthetical expression or a conjunctive adverb at the start of the sentence.

Consider examples below (not the exact sentence but purely punctuation logic):

And[,] also, we will be coming to the party.

And[,] consequently, they failed to produce a report.

But[,] on the other hand, they can offer you advice.

But[,] unfortunately, the government has failed to address our country's economic issues.

Would you put or drop the bracketed comma if the pause is intended after the parenthetical but not after the conjunction? The comma would certainly be there after the parenthetical if the conjunction is dropped and according to both William Strunk, Jr, in Elements of Styles and Gregg Reference Manual, 10th Edition, the comma, in analogues situation, should be dropped after the conjunction in the middle of the sentence if conjunction is starting a 2nd independent clause and is preceded by a parenthetical. See quotes below from each.

Strunk:

If a parenthetic expression is preceded by a conjunction, place the first comma before the conjunction, not after it.

He saw us coming, and unaware that we had learned of his treachery, greeted us with a smile.

Gregg:

Section 142b

When the [transitional] expression or comment occurs at the beginning of the second independent clause in a compound sentence and is preceded by a comma and a coordinating conjunction, use one comma following the expression.

The location of the plant was not easy to reach, and to be honest about it, I wasn't very taken with the people who interviewed me.

The job seemed to have no future, and to tell the truth, the salary was pretty low.

In the first place, I think the budget for the project is unrealistic, and in the second place, the deadlines are almost impossible to meet.

However, in section 126b note, Gregg says the following:

Do not insert a comma directly after the coordinating conjunction unless a parenthetical element begins at that point

I told Calahan that we would not reorder unless he cut his prices by 20 percent And, to my total amazement, he did.

Surely the two guidelines he is giving contradict one another? Why does one need a parenthetical comma in the beginning of the sentence after a conjunction but not in the middle if the only difference is period instead of comma before a coordinating conjunction.

So, according to Gregg, this is correct punctuation:

I told Calahan that we would not reorder unless he cut his prices by 20 percent. And, to my total amazement, he did.

BUT:

I told Calahan that we would not reorder unless he cut his prices by 20 percent, and to my total amazement, he did.

Notice no comma in the second sentence.

You could say that the first parenthetical comma could be dropped on the grounds of prettiness / clarity to avoid excessive punctuation but in this case:

a) It's more logical to drop the comma preceding the coordinating conjunction (and retain the first parenthetical comma)

b) If we were to punctuate the 2 sentences like Gregg advises, then the two sentences read differently as in the first one we are forcing a break / speech pause after a conjunction. And surely, the sentences should not be read differently just because the period before the coordinating conjunction is replaced by a comma?

Is there a grammatical rule on this or in cases like that it's simply a style matter?

I am asking because I personally prefer to drop the first parenthetical comma after a coordinating conjunction on the basis that coordinating conjunction does not have any syntactic role in the clause -- it simply connects the two clauses acting as connector making anything that comes after it an introductory element which would only have one comma in the beginning of the sentence if the conjunction is to be dropped?

Also the natural speech pattern seems to suggest that and the rules above address this situation in the middle of the sentence but not in the beginning?

Any thoughts are greatly appreciated.

A side question: if dropping the first parenthetical comma does turn out to be a mere style preference, would that be acceptable in the formal written works like academic reports and papers (logic being, if the rule is acceptable, it's acceptable in every writing piece).

Answer

As Robusto points out in comments beneath the question, there is no universally acknowledged rule governing whether to include or omit a comma after a conjunction at the beginning of a sentence. Robusto reports preferring to include such commas in academic documents, but many other writers and editors would not include them.

In my experience copyediting manuscripts for book publishers (including university presses) and later for magazine publishers, I don't recall ever having encountered a house style that required adding a comma after "And," "But," or the like. To the contrary, most house styles either said nothing at all on the subject or recommended omitting such commas, presumably for the reason that Words Into Type, third edition (1984) gives at the start of its long section on comma usage:

A comma should be used only if it makes the meaning clearer or enables the reader to grasp the relation of parts more quickly. Intruded commas are worse than omitted ones, but keep in mind at all times that the primary purpose of the comma is to prevent misreading.

The argument for including a comma after an opening conjunction is not, I think, grounded in a desire to make the meaning clearer (since the meaning tends to be quite clear without the comma, as Peter Shor indicates in a comment above), but rather in a desire to demarcate with exactitude the boundaries of the parenthetical expression that follows. Why Gregg Reference Manual would insist on such precision at the beginning of a sentence but not in the middle of one is a mystery to me.

There is nothing inherently wrong with using commas to break out parenthetical phrases regardless of where they appear in a sentence: It increases the number of commas in a work while (arguably) not making the sense of the text any clearer; but it's a style decision, and style decisions—if followed consistently—don't need to be justified.

On the other hand, if you don't want to add a comma after a conjunction at the start of a sentence, I don't think that you should consider yourself to be under any obligation to the preferences of Gregg Reference Manual unless your publisher has instructed you to obey it.