I haven't used Parsedown but I am glad things are moving to CommonMark -- it will shift the focus to performance, structure and extensibility instead of struggling with edge cases and Markdown idiosyncrasies.
PHP CommonMark's compliance is very current--CommonMark spec 0.13, Dec 15, 2014 and it will continue to keep up.
It follows the reference implementation very closely, has good support for extensibility and the code is very well structured. It's likely to become the de facto PHP implementation in my opinion.
It already implements count() and the corresponding interface so it's possible to write count($collection) as if you're writing $collection->count(). It's a valid question why this type of object needs two methods for checking the same property.
Aha! The controversial "Standard Markdown" (is that what it used to be called?) has been renamed to "CommonMark", hopefully avoiding the controversy. (Controversial because Gruber didn't like them using the Markdown name in a standardization effort).
I'm glad. I hope to see implementations continue to spread.
Sorry to add a critic. If you want to make your PHP classes easy to extend, refrain from private and prefer protected. Otherwise, a developer that wants to extend one of your classes will need to also override (i.e. copy and paste) into her class all private members called from the members she really wants to override.
http://parsedown.org/