At my home DZ, we do get hawks that like to flock with and occasionally attack canopies. There's been some damage to canopy cells but not any so substantial as to cause injury. I've had a hawk following me in on final and apparently playing with my pilot chute and topskin, but fortunately no damage.
I have seen birds nearby when deploying. They just went away very fast. One time at Rantoul at the WFFC, I have seen an osprey below me, surprised, curious but friendly. My fear is about big ones like Canada geese and possible impact in freefall or deployment