I have found hay in various horses hay before but not typically my own as I do not board but care for boarded horses. You have every right to be upset, displeased, whatever you would call it. Don’t worry too much about your horse as long as both pieces of twine were found and intact; just keep an eye on her to be sure she is eating and passing manure…if she is, she should be just fine…if she’s not call your vet.
You should expect your horse to be carefully fed and finding both pieces of twine in her hay is certainly not in that category so you absolutely have reason to bring it up to the barn head/manager/whoever is in charge. At my barn, we keep the bale we are working from in a wheel barrow so we can push it from stall to stall; every time we open a new bale, the twine is immediately removed from the wheel barrow and thrown out. You could suggest that some sort of safety precaution like this be used.
-Hope this helps!
The triple threat of riding = EVENTING! 😉