Sounds like a formal art style, but unfortunately it's another term teachers throw at you for bad writing. Parallelism, in short, requires everything in a sentence to match, from tense to number. If I wrote, I will go shopping and made dinner, it doesn't take a trained eye to notice the mistake. To see it clearly, we can split the sentence apart: I will go shopping and I will made dinner. The correct form follows: I will go shopping and make dinner or I went shopping and made dinner. Both work, but imply different actions due to tense alterations.
Parallelism creeps into any kind of sentence, but there are warning signs to look for if prone to breaking parallel structure. Sentences constructed with the following are prone to nonparallel writing: Either . . . or, neither . . . nor, not only . . . but also, anything with both, lists, gerundives and infinitives, tense, and count. This about covers every English sentence. You want every part of your sentence to agree in number (Spaghetti and meatballs is a good dinner option), tense (They will see and conquer), and the many other things I listed above, which are easier taught through example.
Incorrect: We either kill the bird or we let it go.
Correct: We either kill the bird or let it go.
Using We at the start satisfies both parts of the sentence. Otherwise the following is acceptable: Either we kill the bird or we let it go. The same rule applies to neither nor constructs. Think of it as a mathematical setup: We (kill the bird) (let it go). The parentheses indicate the either or situation.
Incorrect: The girls are not only funny, but also they are apt at mixed martial arts.
Correct: The girls are not only funny, but also apt at mixed martial arts.
We have The girls are and get two options: funny and apt at mixed martial arts. Thus we get either The girls are funny or The girls are apt at mixed martial arts. The first, incorrect, example suggests the second construct read: The girls are they are apt at mixed martial arts.
Incorrect: Not only is John strong, but also has a gentle touch.
Correct: Not only is John strong, but he also has a gentle touch.
This time the subject came after the not only construct, thus is part of that phrase, if you will. Using my math method: (John is strong) (he has a gentle touch). We need the pronoun in the second half.
Incorrect: A good man will give money to charity, do community service, and helps out in school.
Correct: A good man will give money to charity, do community service, and help out in school.
Same principle: A good man will (give money) (do community service) (help out). A good man will not helps out. He will help out or won't do anything at all. Good men got good tense.
Incorrect: I wrote this at night, and there is little pride in the results.
Correct: I wrote this at night, and I was not proud of the results.
A little trickier, but should be apparent with the correct version.The second clause of the incorrect example is in passive voice, while the first is in active. Furthermore, the first clause is past tense (I wrote), but the second clause is present (there is). Tense must remain parallel, as must voice. Of course, there are proper exceptions: The movie is funny, which has made it successful.
Incorrect: Good writing consists of many aspects: style, grammar, how to spell, how to edit, and vocabulary.
Correct: Good writing consists of many aspects: style, grammar, spelling, editing, and vocabulary.
Given the correct version, the mistakes should be apparent—we must keep the aspects similar to each other.