With a properly executed power of attorney for finances, you can handle some or all of your parent's finances for them once they're in a nursing home, even if they're still capable of making decisions for themselves. And if the power of attorney is "durable," you can continue to handle their finances once they become incapable of making any decisions.
Using that power of attorney to protect their assets from nursing home costs, though, is the real problem. Once your parents are in a nursing home, there's not a lot you can do to protect their assets. Medicare can cover a short skilled nursing facility stay following a hospitalization. But the only government program that pays for long-term custodial nursing home care is Medicaid. To be eligible for Medicaid, your parents must have very low income and assets. If one parent remains at home, the house and a greater amount of assets could be protected as long as that parent lives there. But once both parents are in a nursing home, they must use up almost all their income and assets to pay for their own care before Medicaid will step in. And if you're thinking of giving away their assets, or distributing them to family members, to get your parent's finances down to Medicaid eligibility limits, be aware that Medicaid has a 60-month "look-back" rule. This means Medicaid considers any assets distributed within the five years prior to applying for Medicaid to be counted as still belonging to your parents.