I guess it depends on the type of parsley. Just regular parsley seems okay, but "Spring Parsley" is bad. Here's a few blurbs I found for just a "parsley bad for cats" search:
Italian and curly parsley and coriander are considered to be non-toxic and edible, but may cause stomach irritation. Mace, nutmeg, paprika and turmeric should be avoided completely.
Many pet supply stores and even grocery stores now sell cat grass. Catnip, sage, thyme and parsley are also safe for your cat to chew in moderate amounts. Wheat (not "wheat grass") and oat-grass sprouts are also safe and tasty for your cat.
Spring Parsley. Toxic to cats, dogs and horses. Symptoms of poisoning include photosensitization (ulcerative and exudative dermatitis) and ocular toxicity.
Give your cat some plants for nibbling, some for sniffing and some for play. For chewing, always keep a pot of tender grass seedlings -- rye, alfalfa and wheat -- growing in a sunny spot. Parsley and thyme are herbs that many cats enjoy smelling and chewing, and both can be grown indoors. Try some different varieties, especially with the parsley.