I sometimes see parentheses around only part of a word. What does this mean? For example, someone typed the phrase "mission (im)possible". I am unsure what the significance is of putting parentheses around (im).
The example you give seems to be trying to sneak an additional meaning in;
The author wants you to read the sentence both ways:
"Saving the firm was a Mission Impossible."
"Saving the firm was a mission (that he saw as) possible with skill/effort."
Here's another example, this time implying criticism of a mad scientist:
"The whole programme was (experi)mental."
meaning the programme was both 'mental' and experimental in the speaker's view.
A(n) set of parenthesis within a word has the same function as using them in a sentence (to provide supplemental information)
In this case, the supplemental information is a dynamic letter.
Generally you see this syntax in technical writing.
Using this parenthesis tip sheet, you can get a broader idea of the use of parentheses.
I don't believe there is an actual word for this kind of thing. Perhaps
To expand on Marc's answer, square (or plain) brackets are sometimes used to "fix" a quote.
For example, if someone were to say:
'I watched Simpsons last night and it was funny episode.'
It might be rewritten as:
'I watched [The] Simpsons last night and it was [a] funny episode.'
The brackets are used in order to denote it as something taken directly from and exactly as another text or statement.