I happened upon a chunk of writing advice on parallelism when using conjunctions by a fellow named Mark Nichol. You can see the actual site here. All of his advice seems pretty fuzzy, like he doesn't exactly know what the heck he's talking about most of the time, so I thought I'd provide some clarity. My comments in red.

1. “We often pay more attention to them than our own children.”
This ambiguous sentence means either that we pay more attention to something than we do to our children, or that we pay more attention to something than our children do. This slight revision reflects that the writer meant the former choice. (“We pay more attention to them” is balanced against “[we pay attention)] to our own children.”): “We often pay more attention to them than [we pay] to our own children.”

Actually, not bad advice, lol. On to the next.

2. “His version is created not with brush and ink, but countless Lego blocks.”
The parallel phrases in this sentence, balanced by the fulcrum but, are not “with brush and ink” and “countless Lego blocks,” but “brush and ink” and “countless Lego blocks,” so repeat with: “His version is created not with brush and ink, but with countless Lego blocks.”

You need not repeat the preposition; it's optional. I don't most of the time. The initial sentence is perfectly acceptable. Also, there is no such thing in grammar as a "fulcrum". In this sentence, "but" is a conjunction, and "not...but" is a correlative conjunction.

3. “The story here is not one of privacy infringement so much as the way real estate is changing because of technology.”
The fulcrum in this sentence is “so much as,” and the phrase “is not one of privacy infringement” must be balanced against one that starts with the same verb: “The story here is not one of privacy infringement so much as it is the way real estate is changing because of technology.”

Uh, no...doesn't need to be the same verb, just A verb. That's the whole idea of parallelism - using the same parts of speech on either side of a conjunction so they mirror each other in structure. The conjunction here is "not...so much as". On one side of the conjunction we have "one of privacy infringement" - a noun, preposition, adjective, noun. We want the same parts of speech on the other side of the conjunction, and "it is the way real estate is changing because of technology" ain't it, brother. I'd probably re-word this sentence. Though I don't know the context, here's an example: "In this instance, the story about real estate is not privacy infringement, so much as technology evolution."

4. “The rainwater boon isn’t so much about taste as reliability in a region where hundreds of wells dried up in the last drought.”
This sentence has the same fulcrum as the previous example does, but notice how the sentence reads more smoothly and has more impact because of the inversion of the constituent phrases: “In a region where hundreds of wells dried up in the last drought, the rainwater boon isn’t so much about taste as it is about reliability.”

I agree. Switching the phrases does make it read better, but he sticks "it is" in there, and it screws up the parallelism. The parallelism was fine in the first sentence.

5. “They protect consumers from purchasing products that are not effective or even dangerous.”
Without the repetition of the phrase “that are,” this sentence crashes to a halt with the false parallel termseffective or dangerous. Omit the first word and the fulcrum from the equation, and the resulting sentence, “They protect consumers from purchasing products that are not even dangerous,” does not retain the meaning. The point about dangerous products needs a complete phrase: “They protect consumers from purchasing products that are not effective or that are even dangerous.”

I don't think this fellow has the slightest clue what he means by "fulcrum". I know I don't. I have no clue what he means by that first sentence, but I think he's trying to say that "not" applies both to "effective" and "dangerous", so why do consumers need to be protected from something that isn't dangerous? How about we jettison the repetition altogether and just replace "not effective" with "ineffective": They protect consumers from purchasing ineffective or even dangerous products.

6. “They believe in cultural and racial diversity, but not diversity of opinions.”
Take away the first phrase, and you’re left with an omission in “They (don’t) believe diversity of opinions,” so the preposition in must accompany both phrases: “They believe in cultural and racial diversity, but not in diversity of opinions.”

How about "They believe in diversity of race and culture, but not diversity of opinion."?

7. “Thanks for your generous assistance and support of these books.”
If “and support” is omitted, the phrase “assistance of these books” stands out as faulty, so repair the error with one of these two options: “Thanks for your generous assistance with and support of these books,” or “Thanks for your generous assistance and for your support of these books.” Better yet, perhaps, is “Thanks for your generous assistance in supporting these books.”

To me this all depends on context. I view "assistance" as probably time and effort, and "support" as probably moolah. While his advice is not technically wrong, I think his last sentence removes the idea of money. I would probably re-word the sentence and state specifically why I'm thanking the person, because the sentence is vague at best. "Thanks for your generous gift of time and money in support of these books.

8. “Beagles rely on their acute sense of smell to chase their quarry and alert hunters with their high-pitched barks.”
Beagles rely on smell to chase their quarry and alert the hunters? No. Their smelling and their barking are two parallel attributes. This sentence requires two independent clauses with parallel subjects: “Beagles rely on their acute sense of smell to chase their quarry, and they alert hunters with their high-pitched barks.” (A fulcrum assisted by a “not only . . . but also” phrase might seem useful at first glance, but that revision alters the writer’s intent.)

Fulcrum...IT'S A CONJUNCTION, FATHEAD! Sheesh. The conjunction here is "and". We want the same grammatical structure on both sides of "and" for good parallelism. So...rely on (verb with a preposition) their (possessive pronoun) acute (adjective) sense of smell (noun phrase) to chase (infinitive) their (possessive pronoun) quarry (noun). I'd probably change both instances of "their" to articles and write something like this: “Beagles rely on an acute sense of smell to chase the quarry and send out high-pitched barks to alert the hunters.”

9. “Those who clashed with the color scheme were getting fired or relegated to the stockroom.”
Without a balance to either side of or, the sentence implies that people were getting fired to the stockroom or relegated to the stockroom. Repeating the verb clarifies that only the second option involved the stockroom: “Those who clashed with the color scheme were getting fired or were relegated to the stockroom.”

No, just no. In what universe would anyone consider "fired to the stockroom" as a possible meaning? The original sentence is okay as it is, but we could improve it. We could replace the empty verb "getting" with "either", so we have a nice correlative conjunction...er, fulcrum: Those who clashed with the color scheme were either fired or relegated to the stockroom.

10. “Families have been leaving the city not so much because of the form housing takes but its price tag.”
The parallel phrases here are (or should be) “because of the form housing takes” and “because of its price tag.” Without the following fix to the second phrase, the reader trips into a prose pothole: “Families have been leaving the city not so much because of the form housing takes but because of its price tag.”

A prose pothole...maybe we could use that fulcrum to get out. Parallelism with conjunctions typically works best with the fewest amount of words. I'd replace "have been leaving" with "leave", get rid of the empty verb "takes", eliminate the repetition: “Families leave the city not so much because of the quality of housing, but the cost.”

I think I'd be extremely wary of taking ANY writing advice from Mr. Nichol.