My experience tells it's either slugs or grasshoppers, albeit I lean very heavily towards the latter. Last year they did a number on the okra and this year it's all out war over parsley.
You can usually tell if it's slugs if you check early in the morning: you will still find traces of slime all over the place. Also most slugs are picky and will just eat the tender leaf tissue, leaving the veins alone. I will post what they did to the eggplants later.
PS: if anyone has a remedy to keep grasshoppers away without covering the parsley I am all ears.