Unless you are concerned about medicaid issues or some other issue that you haven't mentioned, anyone can gift $13,000 to as many people as s/he wants during a calendar year. So I can give $13K per year to you, to your spouse, to each of your 10 kids, to each of your 10 kids' spouses, to each of their 20 kids, and so on.
If the person in the nursing home does not have "capacity" (mentally incompetent), then the person who is their agent under a financial power of attorney can make the gifts provided that the POA form gives them the right to do so. Otherwise, you might need to go to court to get permission to make the gifts.
Obviously, no gifts should be made if it's not in the best interests of the person in the nursing home (at least I hope that's obvious).
The information you obtain at this site is not, nor is it intended to be, legal advice. You should consult an attorney for individual advice regarding your own situation.
Are you asking this question because your parent's nursing home costs are being paid by Medicaid? In Connecticut, Medicaid is also known at Title XIX. A link to the Department of Social Services brochue is posted below.
Medicaid has nothing to do with the gifting amounts discussed in the previous answer.
If you are considering transfering money on behalf of a parent who is in a nursing home, using a power of attorney, read the court case linked below involving a daughter of a nursing home resident who was held responsible for nursing home bills that were incurred after money and property was transferred using a Power of Attorney.
You should consult an elder law attorney in Connecticut who can answer your specific questions. No answer can be given to your question based on the limited information you provided.