US resident alien buying a house?

They should be able to buy a residence anywhere in the United States.

They should qualify for any and all property tax exemptions based on age.

Yes. You can be a co-owner of the house so that the home does not become a probate asset after they both die, but it is not always advisable. However, if either parent has an ownership interest in the house at death, then his or her fractional interest in the home will be included in the determination of the decedent's federal and state estate tax computation. (It should be zero for the first parent to die, there may be an estate tax on the death of the surviving parent.)

Speak to a competent immigration attorney to see if your parents may lawfully purchase a home while their green cards are pending. If so, then they need to speak to a reputable mortgage originator to see if they qualify for a mortgage. This is important unless they plan to pay for the home in cash. Underwriting standards differ, and you don't want them to default in the performance of the Purchase and Sale Agreement and lose their deposit because they applied to a very strict lender.

The other matters can be discussed with an estate planning attorney either before or after they purchase their home.

Good luck.