Adding parsley at the beginning of making chicken stock
It certainly does make sense - adding herbs at the beginning of the process (not just parsley) gives them ample time to infuse their flavor into the stock. I regularly do so along with other herbs: rosemary, thyme, or whatever else is on hand.

This is an optimal place to use up dried herbs (I'm fairly sure a lot of kitchens have a sad, half-used container of dried parsley stuffed in a cupboard; might as well gets some mileage out of it). There's really no need to use fresh parsley when cooking for an extended period, as much of the light, fresh, grassy flavors will be destroyed during the long simmering. In fact, some fresh herbs can even develop a pronounced bitterness after lengthy cooking; I haven't noticed this with parsley, but you might if using a lot of it.

So sure - add your parsley right when you start making your stock, and feel free to use the dried stuff that you can't readily substitute for fresh in other places. The only exception is if you really want to get those lighter, fresher flavors; then there's really no substitute for fresh parsley. You'll have to add it toward the end of cooking, and plenty of it to make sure you get ample infusion.