In clauses with a long subject such as lampooning a bad idea with humor, we do sometimes make a mini-pause in speech before the verb, which is why some pople would place a comma after humor. However this would be incorrect. One of the few absolute rules of English punctuation is that we must not separate a verb from its subject by a single comma.
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.
Best Answer
Bryan Garner, Garner's Modern American Usage, second edition (2003), offers a succinct and reasonable discussion of thus in its various senses from an American perspective:
The sentence that you ask about—
—uses thus in Garner's sense 3, as an equivalent of hence, consequently, or therefore. But instead of appearing at the beginning of a major clause, it appears in the midst of the phrase "new and better," where both new and better are adjectives modifying the same noun (solutions); in such a position, Garner implies, thus should not have a comma after it. According to Garner's analysis, the case would be different if the sentence read as follows:
Of course, the meaning of that sentence differs significantly from the meaning of the sentence that you ask about. But while we're on the subject of meaning, I must echo the observation in ugajin's answer that including thus in your original sentence makes sense only if you truly intend to say that, simply by virtue of being new, new solutions are better than older ones; as a matter of logic, that's a problematic assertion to make, since it's easy to imagine new attempted solutions that turn out to be worse than the status quo.
In my view, whether a comma is appropriate after thus even at the beginning of a clause is ultimately a matter of style preference; but I agree with Garner's implicit view that, when thus appears in the middle of a clause, it normally does better without a comma.