As residents of a state, taxpayers must report and pay income tax on all their world-wide income that they receive while a resident of the state.
Residency is determined by your domicile. Your domicile is your permanent, "true" home. You can only have one domicile at a time, even though you might have several homes you stay in. Domicile is determined by analysis of all relevant factors and it includes things like home ownership or lease, but also where your closest family and friends are located, your doctor and dentist, significant social relationships like your church and blowing league, voter and vehicle registration, and so on. When you change domiciles, you must take active steps to establish a new domicile AND you must take active steps to abandon your former domicile.
You can read more about California's take here.
https://www.ftb.ca.gov/file/personal/residency-status/part-year-and-nonresident.html
Your parents would owe California income tax on the sale of their home if they closed on the sale after establishing their domicile in California.
To minimize the risk of California being able to claim that your parents established a new domicile before the sale, your parents should probably not only not move to California, but should not sign a lease, close on a house or change their voter and car registrations until after the Washington house is sold.
HOWEVER,
California follows the Federal rules for excluding capital gains on the sale of your main home. If your parents have owned their present home at least 2 years, and lived in it as their main home for at least 2 of the past 5 years, they can exclude the first $500,000 of capital gains from taxation. If their gain is less than $500,000, it would not be taxable in California even if they did not close on the sale until after they moved.
https://www.ftb.ca.gov/file/personal/income-types/income-from-the-sale-of-your-home.html
