Does anybody know if I can change my work hours to part time and get Family and Medical Leave Act to make up missing hours?

IMPearl, I know some employees will take FMLA to help out their aging parents but I wouldn't do it unless it was a dire emergency. Use vacation days, sick leave days, and days without pay. It was a struggle as I had a corporate manager who was not very understanding whenever I asked her for an hour here or there off to take my parents to appointments, of which there were many.

Thank goodness I didn't use one hour of my FMLA for my parents [there were no emergencies] as out of the blue I developed a very serious illness to which I was glad I had my 12 weeks of FMLA to use for my recovery. I was lucky enough that my employer had a separate insurance plan for their employees that would pay some of my salary during that time frame [something similar to AFLAC]. It was half my salary which I welcomed. Plus I was able to keep all my benefits. It all depends on your employer.

If your Mom has severe dementia, chances are she would need 3 shift of caregivers each day. Could she pay from her own retirement for that care? Could she afford to live in a memory care center?

IMPearl, You need to know FMLA does NOT make up the slack. It is Unpaid leave, not part-time, no benefits. Read your company's FMLA Plan.

I think the key to your question is pay. FMLA does not provide pay itself. That is largely dependent on your employer. Some employers will allow you to draw on accrued vacation/sick/personal days etc. for compensation during the time you are gone. What FMLA does do for the allowed period of time is ensure that your position will be there for you when you return to work.

Info from the Department of Labor website FAQ's on FMLA
(Q) What does the Family and Medical leave act provide?
The Family and Medical Leave Act (FMLA) provides eligible employees up to 12 workweeks of unpaid leave a year, and requires group health benefits to be maintained during the leave as if employees continued to work instead of taking leave. Employees are also entitled to return to their same or an equivalent job at the end of their FMLA leave.
The FMLA also provides certain military family leave entitlements. Eligible employees may take FMLA leave for specified reasons related to certain military deployments of their family members. Additionally, they may take up to 26 weeks of FMLA leave in a single 12-month period to care for a covered servicemember with a serious injury or illness.
Coverage
(Q) What types of businesses/employers does the FMLA apply to?
The FMLA applies to all:
public agencies, including local, State, and Federal employers, and local education agencies (schools); and
private sector employers who employ 50 or more employees for at least 20 workweeks in the current or preceding calendar year – including joint employers and successors of covered employers.
Eligibility
(Q) Who can take FMLA leave?
In order to be eligible to take leave under the FMLA, an employee must:
work for a covered employer;
have worked 1,250 hours during the 12 months prior to the start of leave; (special hours of service rules apply to airline flight crew members)
work at a location where the employer has 50 or more employees within 75 miles; and
have worked for the employer for 12 months. The 12 months of employment are not required to be consecutive in order for the employee to qualify for FMLA leave. In general, only employment within seven years is counted unless the break in service is (1) due to an employee’s fulfillment of military obligations, or (2) governed by a collective bargaining agreement or other written agreement.

There are both federal and state family leave laws and some depend on the size of your employer - best to check with your employers' HR Dept assuming there is one- if not do a little research on the internet starting with a search for your state

Caution though that I believe that federal family leave is only for 12-weeks every 12-months not every year and that it is unpaid leave or perhaps 50% replacement pay

I too have considered going part-time but at 55 I still do not have enough saved for retirement and so I'm pushing that out and am saving the family leave for the real emergency - I have been doing this for nearly 9-years and have managed all the falls and various trips to the ER and hospital stays with sick time and vacation time - fortunate to have had an understanding boss but he just retired ....

At some point your mom will require constant supervision not just while you are at work -

You have a key thought there: Bills have to be paid. The physician was a bit out of line telling you that you needed to spend more time with her. Maybe you should ask him to supplement your income.

Dementia can last for years even in the later stages. Did the doctor give any prognosis on the time your mother has left. FMLA won't cover long term. Tell us a bit more and someone could have some good advice about what to do.