Parsley, curly parsley or garden parsley is a species native to the central Mediterranean region and has been naturalised elsewhere in Europe. It is a widely cultivated herb, spice and vegetable.
It is widely used as a garnish and is also widely used I. European, Middle Eastern and american cuisine.
So can rabbits eat parsley at all?
Lets take a look at their nutritional data in particular their calcium, sugar, phosphorus, acidic, fibre, fat and sodium content is most pertinent as far as piggies are concerned.
Nutritional value per 100 g (3.5 oz)
Energy 151 kJ (36 kcal)
Carbohydrates
6.33 g
Sugars 0.85 g
Dietary fiber 3.3 g
Fat
0.79 g
Protein
2.97 g
Vitamins
Vitamin A equiv.
beta-carotene
lutein zeaxanthin
(53%)421 μg
(47%)5054 μg
5561 μg
Thiamine (B1) (7%)0.086 mg
Riboflavin (B2) (8%)0.09 mg
Niacin (B3) (9%)1.313 mg
Pantothenic acid (B5)
(8%)0.4 mg
Vitamin B6 (7%)0.09 mg
Folate (B9) (38%)152 μg
Vitamin C (160%)133 mg
Vitamin E (5%)0.75 mg
Vitamin K (1562%)1640 μg
Trace metals
Calcium (14%)138 mg
Iron (48%)6.2 mg
Magnesium (14%)50 mg
Manganese (8%)0.16 mg
Phosphorus (8%)58 mg
Potassium (12%)554 mg
Sodium (4%)56 mg
Zinc (11%)1.07 mg
Link to USDA Database entry
source wikipedia
As you can see parsley contains a lot of phosphorus, calcium, is very acidic, and contains a hint of sugar, fat and fibre.
This means that rabbits cannot eat parsley. There is too much acidic, calcium and phosphorus content which is bad for them and it will make them ill if they eat it.
If they nibble at it they should be fine but anything more that that is not good.