Asparagus isn't toxic, but it's a debated vegetable — some birds get gas or digestive upset from it, and many simply dislike it. Offer a small cooked amount and see.
Keeper's noteLightly steam it to soften, offer a small piece, and watch how your bird responds. If it causes loose droppings, drop it — there are easier greens.
What to watch out for with asparagus
There's no toxin in asparagus, and it brings vitamins A, C, and K. But it contains compounds (and a lot of fiber) that can cause gas or loose droppings in some parrots, and raw stalks are tough. It sits in the 'fine for some birds, not worth forcing' category rather than being a clear win.