Sometimes they're interchangeable, sometimes they're not.
So let's finish with commas. They're kind of dull, which means you should always consider using them because punctuation usually shouldn't be drawing attention to itself. There are probably a hundred different rules that govern how to use commas, so I'm going to limit this discussion to commas that could be used like parentheses or dashes.
Commas don't interrupt your sentence, so you use them when the words you're enclosing are a natural part of your sentence and not some comment from left field or flamboyant statement. Commas are generally used for appositives, for example, which are defining or clarifying statements after nouns. Here's an example of an appositive set off with commas:
My youngest sister, Meghan, will be visiting soon.
"Meghan" just tells you who my youngest sister is. You could set her name off with dashes as we did in the earlier sentence about George the accountant, or with parentheses like we did with a date earlier, but there's no reason to in a sentence like this one.
Commas are also used to set off non-restrictive elements such as "which" clauses.
Diamonds, which are expensive, aren't something I buy very often.
Like a parenthetical, the "which" clause could be left out of the sentence without changing the meaning. I actually did a whole episode just about "which" versus "that" and commas, so you can read if you'd like to learn more.
As I'm sure you've noticed by now, you could make a legitimate argument for using at least two different punctuation marks in nearly every example sentence I've given you, but these general rules may be helpful:
Use parentheses when you want to enclose something that is incidental to the sentence, something that is background or almost unnecessary.
Use dashes when you want to enclose or set off something that deserves a lot of attention, is meant to interrupt your sentence, or already has commas or parentheses in it.
Use commas to enclose things that belong firmly in the flow of your sentence.
I know it can be frustrating that there aren't hard-and-fast rules about when to use commas, parentheses, or dashes, but learning to use your judgment is part of finding your voice and becoming a better writer. In this case, the rules are more like the pirates' code in Pirates of the Caribbean—they're more what you'd call guidelines than actual rules. I bet Jack Sparrow never used a comma; he seems like a dash man to me.
The Grammar Devotional
I'm Mignon Fogarty—author of The Grammar Devotional, which makes a fantastic graduation gift.
Parenthesis image courtesy of Shutterstock