@Mark G.
Yes, there is a way to transfer property from a parent to a child and keep the assessed basis for property tax purposes the same. You need to file paperwork with the county though to stop it from being reassessed and alert them it is a parent-child transfer. It should not matter that the property is a rental property as the exclusion is available for both primary residence or other properties. It will possibly matter though how the property is titled or held, if in any entities.
The fact that you're talking about buying it at below market rates, is a potential problem though. This may have to be recorded as a part sale-part gift then if your parents are not going to charge full fair market value. The difference of fair market value versus what you pay would likely be considered a gift to you, which may require the filing of a gift tax return, and would eat into their lifetime exclusion depending on the amount of the gift. If you do a non-interest note, that also would be considered either a gift or your parents would have imputed income for the amount of the interest they're not charging you. Everything needs to be done at "arms length" for it to be on the up-and-up. Potentially your parents could charge interest but forgive it each year using their annual gift tax exclusion. The interest rate doesn't necessarily have to be market rates, though. The IRS lays out a guideline for family loans which I think is the AFR, which is usually below the market rates.
My understanding with regard to 1031 transactions is that the party needs to be the same on both sides. So if they enter into the 1031 as an individual, they need to purchase the new property as an individual as well. They should eventually be able to contribute to an LLC though.
Could your parents just gift you the property outright? How old are they and what do you expect their relative life expectancy to be? What's their relative net worth? Under the newest tax laws that just came into play in December, an individual can gift or die with a combined $11.18M before being subject to transfer taxes (and $22M per couple with portability or split gifts). That law is supposed to be in effect for the next 10 years, indexed upwards for inflation. So if you don't think they will have $22m combined by the time they die, they could just gift the entire property to you, and pay no taxes. If you receive the property by gift, however, you will take a carryover basis so you will be subject to large capital gains when you eventually sell it. To avoid the capital gains taxes, they would have to hold it until their deaths so that you receive it with a stepped up basis via inheritance. Also note that if you were to purchase the property from them, your purchase price would become YOUR basis in the property and if you're paying below market prices, then you have a built in gain as well that would be recognized when you sell it if you sell for more than what you paid for it.
I have the names of CPAs/attorneys if you need, though they're mostly in San Diego and Southern California.
*None of this post creates an attorney-client relationship or CPA-client relationship. It is not to be relied upon and readers are advised to seek professional advice.
