The tortoisetable website is great but super ultra cautious. Variety of foods is the key, in my opinion. If you give parsley, or anything else for that matter, every day for months, years - like the old days when tortoises only got lettuce - then there will be problems and lots of them and early death. Parsley every now and then, is okay, me-thinks. Parsley has a lot of medicinal values. It was first used as medicine before it was used as food in Roman times. It has Vitamin C and it is very anti-inflammatory and I have heard that it can draw out toxins and even unwanted metals, like mercury, in the body. Hopefully, neither you nor your tortoise have that but the point is that every now and then, it should be okay. I make parsley tea for me and my pets get the parsley afterwards mixed in their foods. Sometimes it's my dogs, sometimes it's my tortoises. Nary a problem. I learned this from my grandpa, and his desert tortoises which our family inherited when he died, are over 60 years old, raised from hatchlings he found in the desert (when there were no laws against). They have had a variety of plant/green things, including parsley, over the years. Hydration and water always available is also key so that if there is anything that needs to flush out, it flushes out with the water source.
I've offered parsley in the past, but none of my torts like it.
It's a good plant to grow in enclosures if you want something that won't get gobbled up immediately.