You'd have to eat a LOT of it for your supply to be affected. The small amounts used for culinary applications will do no harm to your milk. Just don't drink parsley smoothies.
Originally Posted by Midwitch
I'm pretty sure you can eat it without any problems, as long as it is not every day. It is true that parsley can be used to lower/stop milk supply, but in order to do so you have consume large amounts, like drinking parsley tea on a daily basis. The way it is used in cooking and salads, is no problem.