Identify the sentence(s) that use parentheses correctly.
A) My sister (and I) both were invited to their wedding.
B) April (and her dog sam) are always welcome to our home.
C) The show will be aired at 8 p.m. (IST) on ABC news.
D) After a week of shooting, the crew was tired of his mannerisms (or rather, lack of it).

Hint: Parentheses ( ) are a punctuation mark that is used to enclose additional or extra information in a sentence without affecting its grammatical structure (a single bracket is called parenthesis, while a pair is called parentheses). The sentence will not be affected grammatically if the words enclosed in parentheses are omitted.

Complete answer:
Now let us analyze the given options:
Option A) My sister (and I) both were invited to their wedding : This option is incorrect because when a phrase "and I" is removed from a sentence, the resulting sentence will sound incomplete, as "both" can't refer to the sister alone.

Option B) April (and her dog sam) are always welcome to our home : This option is incorrect because April is one person and "are" is used for plural nouns if it is removed, the sentence will sound incomplete.

Option C) The show will be aired at 8 p.m. (IST) on ABC news : In this sentence, the information in parentheses ``IST" is additional. IST is the time zone and time zones are enclosed in the parentheses. Hence, this sentence uses parentheses correctly.

Option D) After a week of shooting, the crew was tired of his mannerisms (or rather, lack of it) : In this sentence, the information in parentheses "or rather, lack of it" is additional. "or rather, lack of it" is extra, when omitted the sentence is not affected and is still complete. Hence, this sentence uses parentheses correctly.

Thus, the correct answers are option ‘C’ and ‘D’.

Note: A parenthesis is a punctuation mark that is used to insert information into a sentence's logical flow without disrupting the original structure. To round out the reader's experience, they may provide informative descriptions, editorial commentary, or even a funny word picture. When this happens, pay close attention to what's said inside those little 'bracket things'—it makes all the difference.