Yes, in small amounts — a sprig of parsley now and then is fine. Very large quantities have been linked to photosensitivity in some birds.
Keeper's noteA sprig in the chop mix or clipped to the bars occasionally is no problem. There are herbs with better résumés though — basil and cilantro can be offered freely.
What to watch out for with parsley
Parsley contains psoralens, compounds that in large doses can cause photosensitivity (sun-triggered skin reactions) — documented mainly in livestock that grazed on huge amounts. The doses in normal sharing are far below that, but it earns parsley a 'moderation' label rather than a free pass.
How to serve it
A sprig or two occasionally, washed well.
Mix into chop rather than offering as a daily green.
Prefer basil, cilantro, or dill for unrestricted herb foraging.