Take a look at this meaning:
By the way, I marked position 2854 (in the code. It's the slash).
Is is correct to start a new sentence within a parenthesis like this? Or another, worse, example:
By the way, I marked (a lot of things of which you are unaware. Please try not to eat) the semicolon.
If it's not possible to do this, is there than any other solution (except rephrasing!).
Your first example is easily changed:
By the way, I marked position 2854 (in the code; it's the slash).
A semicolon is used to join two distinct sentences that you want to be related. Since the second is in the parenthetical, it's obviously closely related to the first. So it's even better with a semicolon.
Consider the fact that parentheses are used for optional information. If you removed the parenthetical, does the sentence have the same meaning? If yes, you're using it correctly, and you should also consider removing it entirely. If "It's the slash" is essential information, the sentence should be:
By the way, I marked position 2854 (in the code). It's the slash.
Now, as for your second example: It's an incorrect use of parentheses, regardless of the use of two sentences. As noted above, a sentence without the parenthetical should have the same meaning. "I marked the semicolon" does not mean anything like "Please try not to eat the semicolon". In fact, the parenthetical qualifies something that comes before it. It's not connected to what comes after, so you cannot continue a sentence started inside the parenthetical. It would have to end inside. To summarize: This usage is 100% wrong and confusing.
Something like the following is also wrong:
That's something (a bear. A bear) with no fur.
The meaning is basically the same both with and without the parenthetical, but consider what happens when you try to split up the sentences.
That's something (a bear.
A bear) with no fur.
That's obviously nonsense. If they were truly separate, fully-formed ideas that should be separate sentences, then you should kill the parentheses and actually make them separate sentences:
That's something. A bear with no fur.
Again, a semicolon works best when you're truly conveying one idea.
That's something (a bear; a bear with no fur).
Of course, the example is silly, but it's grammatically correct!
It's not correct. It may sometimes be effective and humorous, but that's a different matter.
The first one I'd rephrase by removing the parentheses. They don't add anything there except confusion.
The second one I'd leave as it is if it's an informal piece of writing, on the grounds that it's fairly humorous and works. This is one of those cases where you can knowingly break the rules to create a particular effect. In formal writing, the parentheses and everything between them should be deleted, because the effect doesn't work in such a setting.
I was taught to avoid using parentheses as much as possible. As a rule that's not entirely reasonable (so ignore it), but it did break me of the habit of using parentheses all the time (because I could). A lot of the time you can bracket your aside with commas instead, like this, and the result sometimes flows better. If your sentence looks like a confusing mass of commas when you do this, you probably needed to simplify it anyway.
It is not correct punctuation to start a new sentence within parentheses. The text within the parentheses should be able to be removed without altering the semantics of the full word sequence.
To test:
By the way, I marked ... the semicolon.
though grammatical is not what you meant at all.
You may have really wanted to say it as:
By the way, I marked (a lot of things of which you are unaware). Please try not to eat the semicolon.
or
By the way, I marked a lot of things (of which you are unaware). Please try not to eat the semicolon.
Either way, the sentence must 'close' and if the period is inside the parentheses, it can't. Also, in this example, I don't see the point (or grammaticality) of parenthesizing "Please try not to eat". Also, why did you want to parenthesize the parts of two separate sentences together (what did you expect that to impart? What is parenthetical/an aside about that particular sequence?) ?
