Throughout the last 40+ years, I've had clients whose elderly parents transferred asses into their name to avoid probate, or for Medicaid or nursing home planning purposes, or just because they thought it was the smart thing to do. Unfortunately, this move ended up costing them a serious amount of money in capital gains tax when they sold those assets.
When someone dies, their heirs inherit their assets with a "step up" in basis. This means that the cost (basis) of the asset is what it was worth on the day the person died. But, if you receive a gift from someone (not an inheritence) your cost (basis) is the same as the cost (basis) of the person that gave it to you. If your parents lived in their home for decades, or if they have an investment account that they built up over many years, their cost (basis) in those assets can be a fraction of what they are worth on the date of death. So, when you sell those assets that were given to you (not inhertited) you will pay capital gains tax based on your parent's cost (basis) and not what it was worth on their date of death..
There are ways to remove assets from a person's name and still maintain the "step up" in cost (basis) through the use of a trust. Or, if your parents simply transferred their assets to you (and they are still alive) you can normally undo this transfer to preserve the "step up" in cost (basis). Hire a lawyer who is familair with trusts to make sure this is done correctly.