i think that's where the original purchase price would be the issue. I'm in California and here property taxes are set at 1% of the sales price.
there are some exclusions on this for sales to one's children. I think it's prop 58. it allows a child purchasing a parent's home to choose to either use the current appraised value under prop 13 rules OR their parent's basis. depending on the value of the home at transfer it sometimes works better for the new owner to take their own prop 13 basis (especially if parent's home went down in value and they never applied to get it lowered/taxes lowered).
had a friend who purchased a home from his dad who had owned it since the 60's-home was valued hundreds of thousands more than dad purchased it for. friend took advantage of the parent to child sale law such that he saved tens of thousands in taxes over the years he owned it.