It's time to get your dad on Medi-Cal. This is the California Medicaid program, and your parents' assets are within 20K of the limits. Your mom (the “well spouse”), is allowed to keep $148,620 in countable assets. Your dad (the Medi-Cal spouse) is allowed $130,000. That's very close to the $300K they have in stocks.* And "property used as a home is exempt (not counted in determining eligibility for Medi-Cal)." So your mom can stay and live in the house regardless of whether your dad has to go to a facility. Oh, and they can keep their pension "without having to contribute it all to Share of Cost". For completeness, your mom can keep *all* of any pension in her own name, and if that income is less than $3,716 (which it is), she can keep up to $3,716 of your dad's income.
See here for details/examples for supporting the well spouse:
https://canhr.org/wp-content/uploads/FS_MEDICAL_Spousal_Impoverishment_HCBS.pdfDon't sell the house before applying your dad for Medi-Cal, as then it becomes cash. This site:
https://canhr.org/your-home-and-medi-cal/ suggests you could at some point place the house in your mom's name only after dad is accepted to Medi-Cal and then she'd be able to sell it.
*What to do with $20K? It's less than 3 months of cash, so by the time Medi-Cal is approved maybe it'll be gone? If not, they could do projects on the house, replacing things that are close to their life expectancy like roofs, water heaters, or furnaces, maybe adding some insulation. Whatever you do, don't hide it with a transfer to your own account. That's a big no no.
Getting your dad on Medi-Cal should greatly enhance your mom's ability to cover life expenses. Life expectancy at 82 is 6.77 years for men, and 7.98 years for women, so your parents have many years ahead of them still. Without Medi-Cal they might run our of funds:
Your parents' pension only covers 36K of their 120K spend. You can get 40% to 60% of their home value in a reverse mortgage, so that's 200K to 300K. They have an additional 300K in stocks. At their current spend rate (and your dad's care costs are increasing), these funds (stocks and reverse mortgage funds) would be exhausted in the next 6 to 7 years. Selling the house instead could get them 9.5 years, but some of that will be undone by greater spending on housing.