My siblings went thru this on our parents family home in 2002 in the Bay Area. They still pay on the Prop 13 tax assessment. When the title changed, the county reassesed the house and upped the property taxes. At that time, they were told to pay the increased taxes and then file some additional paperwork with the county to certify they were son and daughter of the original owners. After the county reviewed the additional paperwork, the value was returned to the original assessment and the additional taxes paid during the interim was refunded. IIRC, one of the later Propositions that modified the original Prop 13 included this change.