This response will be strictly with respect to American rules, as I am only familiar with them (though I do understand Brits do this differently).
The rule you want in the Chicago Manual of Style is 6.111 in the current edition (16th), which directs you to all the relevant rules with respect to "quotation marks relative to other punctuation and text."
You never want to have a period both inside and outside the closing parenthesis. (By the way, in the subject of your post, you should say "parentheses," which is the plural, not "parenthesis," which is singular.) If your parenthetical expression is a full sentence (as my previous sentence is), put the period inside. If it is not (as in my first paragraph) the period goes outside.
With question marks and exclamation points, the rule has to do with what is being asked or exclaimed. If the quotation is the exclamation, the exclamation mark goes inside. If the sentence into which the quotation is inserted is the exclamation, the mark goes outside. Same with question marks.
Admittedly this can get complicated in certain circumstances, which is why Chicago has quite a lengthy discussion of this issue.
In the example you've given us:
I know if a sentence is inside either, the punctuation is inside (I also know I use a lot of comma splices. I think of the way the sentences sound in my head and break them up that way.).
You've actually got a full sentence preceding your parenthetical expression. I would do this:
I know that if a sentence is inside either parentheses or quotation marks, other punctuation goes inside them as well. (I also know I use a lot of comma splices. I think of the way the sentences sound in my head and break them up that way.)
It's also not immediately clear what "either" refers to as you originally wrote your sentence, so you should insert your nouns.