Parsnips are fine on Green and Extra Easy days. They need to be synned on Red days, though.
I think perhaps you're getting confused between free and superfree?
Superfree foods are fruit and veg that are free on all three plans - Red, Green and Extra Easy. Because peas and parsnips have to be synned on Red days, they are not superfree, and therefore can't count towards your 1/3 superfree on EE.
However, you can still have them as free foods on Green and EE, just make up your third superfree in other ways!
The vast majority of fruit and veg is superfree. The exceptions are peas & sweetcorn and starchy veg like potatoes, sweet potatoes and parsnips. These would all be synned on Red and ONLY Red days.
Just to add more confusion, you can replace the peas and sweetcorn with sugarsnap peas, mangetout or babycorn, as they are superfree and therefore free on all plans. Starchy veg can be replaced with things like swede, butternut squash, celeriac, and so on.
Hope that's helped and hasn't confused more....!