Does living rent free count as my pay?

No. Living rent free does not count as pay. It counts as slavery or indentured servitude and neither all allowed anymore.

The minimum pay for any job in this country is minimum wage. Live-ins do not get paid by the hour though. Live-ins are also supposed to get time off. A day off for a live-in is 24 hours. That means they can go and do whatever they like and someone else stays with the client.

Why would you or the other caregiver ever have agreed to a job that has no wages? Or were the two of you expecting pay and never received it? If such is the case get the son in trouble with the state of California and walk away from this job.
My friend, let me give you a good piece of advice if you're planning on staying in the field of in-home caregiving. I apologize if it's long-winded.
Never allow the wages to ever be paid late or short. There must be the Understanding between the caregiver and the people who pay him/her. The Understanding are rules agreed upon with the caregiver and the employer. The rules do not even have to be in writing if you're getting paid in cash. They give their word and you give yours.
The Understanding I've always had with any client or family I took work with is this.

1) I work a schedule. There is no staying later or visiting or anything else. If I agree to cover for another caregiver on the same job then fine. If there's an emergency and I'm available then I'll agree to go to the client after my hours. I always get paid for it.

2) I do not take calls from clients or their families after hours. I let them all go to voicemail. If someone was making a cancellation or it's an emergency, I'd call them right back. I always made myself very clear that I don't take client calls after hours because they just want to talk. I will not take a call from a family member asking me if I could "just go and check on them". Or asking if I'll just talk to the client on the phone because they're bored or lonely.
It's the job of the family to check on them after hours or to take their calls when they're lonely and bored.

3) The Pay. Every care job I've ever worked the people responsible for paying knew they need to always make sure I'm paid on time, in the full amount with no exceptions. Every Friday there'd be a receipt for whoever was paying me with my dates of the week, my hours, and the wages I'm owed. Clients able to pay me themselves did so at the end of my week in person. If a family member was doing the paying, often they'd leave my check and we worked on an honor system that way. Or I'd leave my receipt and a check would be left in its place when I came Monday morning to work. Naturally in the course of many years doing in-home care, sometimes on private jobs something would come up and my pay wouldn't be there. Usually the people paying me would call to explain and apologize. I allowed a one time, two-day grace period for them to pay me in full. It was understood if anyone tried to get cute or stiff me on my money, that I'd get in my car and drive away, leaving their "loved one" to fend for themselves. I never had to make good on that threat. The threat alone was enough and I always got paid. In private care jobs when people know you mean business, they treat you as such. Don't ever let any client or family take advantage of you even one time because it will never just be one time. It will be every time.

You are a domestic slave. You should be getting paid as well as free room and board. Your duties should be spelled out in a contract. You should have "time off." Please notify the authorities. When you do, the woman you cared for will be placed in a facility and you should receive help from social services to find another living and working option.

My dad has live-in caregiver. Paid $3500 a month. Yes you should get paid. Get a notebook and write down everything you do for this poor woman. Date time activity etc. Do you do grocery shopping? Bring to doctors? Record all!! Laundry? Cleaning house? Attach a monetary value to each episode. This does not equal free rent. Get out as soon as you can!!

Yes, you should be getting paid. I'm shocked that you aren't be paid. When I looked into round-the-clock care for my mother, it was $11K per month, believe it or not, about the same as a nursing home. Four different people would stay with her in 6 hour shifts at $16/hour. And that was three years ago.

I would think that at least something, say, $3000 a month would be fair. When this woman passes, you will have nothing to show for your time in caring for her -- unless you are independently wealthy. If she passes tomorrow, where will you stay and how will you find the funds to move on?

NO! NO! NO!
It is not legal to have you work for no pay.
I hope that the caregiver that was hired and quit because of no pay is talking to the Labor Board, or a Labor Attorney.
You can also not work 24/7 you need breaks and days off. That is the law.
Since you have access to a computer search the state labor laws on how many hours and days you can work. And there should be information on a live in situation.
4 -5 years ago when I was paying caregivers I paid them $12.00 an hour.
Taxes were taken out and reported.
(I think at this time I would probably pay $15.00 but I would have to do a bit of searching to be sure, this is just off the top of my head.)
I hope your "employer" is properly reporting your wages so it does not later effect your Social Security AND any reporting is done so that the IRS does not come after you later. That is IF you are getting any payment at all.
A quick search I found this.
Live in caregiver wages are between $12.00 and $25.00 per HOUR
Another source Average pay is $687 per WEEK.
And you can not work 24 hours you need time off and if you work more than the legal number of hours (check your state) you MUST be paid overtime.

PLEASE do not let this family abuse you any more PLEASE talk to a Labor Attorney or go to your Labor Board.

Let me get this straight. You are working for free for someone who is NOT your family member? You made an agreement to move in and in exchange for rent you are there 24/7, in charge of all care and of hiring and firing, but you are not employed by the family of this 95 year old?
There is something strange about this picture. My fear is that this person will die, and you not only will have ZERO money, but also no job history, no money and will be homeless. We actually often see this when family members take on this kind of care. The house goes to medicaid clawback and they go to a homeless shelter.
Do follow the advice in some comments below. As you have lived here even albeit rent free and caring for someone you can under California law, at least, be considered a tenant, so it will not be easy to kick you out without tenant relocation stipend of some amount, but otherwise, upon the death of the person you are providing care for voluntarily, you will be on your own. I am wishing you good luck in addressing this, but it is 3 years after the fact; short of serving notice and moving I cannot imagine what you can do.

If I was you I would be making alternate arrangements as this is slave labour if she was to die tomorrow where does that leave you?

No one should work 24 hours a day and 7 days a week.
Especially if you are not being paid. I doubt your situation is legal.
Please, please do not continue to do this, you are being treated like a slave.
You need money for your own life, and future
This is so upsetting to read, please look after yourself

24/7 labor in exchange for room and board is slavery, which is frowned upon in most societies.

Some napkin math says you've provided over $300,000 worth of free labor. Start charging ASAP or get a job. You'll be on the streets if this lady drops dead tomorrow unless you're wealthy.

I pay an agency $96 for four hours ($24 an hour) of care twice a week for my husband who has dementia. She does cognitive therapy with him including puzzles, play cards, etc., no household chores.

The agency I use charges $340 for 24 hours. The caregiver must have eight hours of free time during the 24 (so she can rest or do whatever she wants).

Twenty years ago, my mom and dad paid their live-in caregiver $120 a day plus room and board.

As others have said, you need to be paid!!!!