Since you never see the down and it’s kept safe within the nylon shell, most folks will be okay to use it even with pet dander allergies. Occasionally, some down may escape as your jacket is used and you may be exposed to the feathers. If you are concerned, check with your doctor or allergist.
There are non-down jackets, or synthetic jackets, that try to emulate the same benefits and insulation. Our main focus wasn't on these, but Primaloft and other synthetic insulation options can still provide decent insulation, but down is king.