How can estate planning help to preserve some of an elderly person's nest egg from nursing home costs?

isn't what they worked so hard for all their lives is to provide for THEIR retirement, their "golden years" their care comes first.
Would you rather have their assets squirreled away and them in a average, below average facility (Memory Care or Skilled Nursing) or find a good facility that later will accept Medicaid AFTER they have private paid for X number of years.?
Unless they are very wealthy or are in very poor health no matter what you try to do to "protect assets" they will probably outlive what they have saved.
I have a problem with "trying to protect assets" for heirs while applying for Medicaid and letting taxpayers cover the cost of care. My opinion is they earned it, they saved it they are entitled to get the best care that their money can buy for as long as it lasts.

I always become a little concerned if I see the word “protect/preserve” in a question like this.

If you are thinking along these lines with respect to ensuring that you parents receive nothing but the very best care as they move into elder life, I congratulate you for your wisdom and compassion.

If you are thinking about protecting the estate they will be leaving behind to their survivors, please consider who “worked so hard”, and who may need care that can be expensive and difficult to provide.

My father worked his whole life and my mother was a shrewd and confident manager.

When my father died without leaving a will, I was entitled by law to a third of his assets, and I unhesitatingly turned my share over to my mother, because I totally believed that it was rightly hers.

Over 20 years later when she entered a superior SNF 10 minutes drive from my house, I wound up paying almost $1,000,000 OF HER MONEY on her care.

Your perception of high quality, compassionate SN care may be different from mine, but I came to learn that Karma can enter our decision making, for the right reasons.

Seek out a good decent honest lawyer skilled in your state laws regarding financing elder care. Perhaps you will find that your current assessment of pricing in AL/SN care is not what your question is suggesting.

May your decisions be loving ones.

Contact an elder care attorney. While it may be expensive, they will help you “spend down” and “move” around money the best way possible without raising red flags.