There is a lot of weight in the parsley stems and it is therefore tempting to include them in the chimichurri.
I have also seen their use suggested on some sites.
What are the pros and cons?
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Depending on how thick the stems are, they might get quite hard and taste "wooden". They also get more bitter the more you travel down the stem. So, don't be super picky when discarding stems, but don't be too generous either. I usually discard stems when they start to get somewhat rigid when you try to bend them.
Here's an article I quickly found on the subject matter.