According to IRS publication 926 and you are correct - you should not count wages you pay to your parent as social security and Medicare wages, even if these wages are $1,500 or more during the year unless both the following conditions apply.
Your parent cares for your child who is either of the following.
Your marital status is one of the following.
However these wages are taxable income for your parent and should be reported to IRS. You need to submit W-2 form that would report zero in SS and Medicare wages fields.
This type of wages is not subject to FICA taxes, it is not a subject of FUTA tax - please also see Circular E page 8 - http://www.irs.gov/pub/irs-pdf/p15.pdf - so when issue W-2 form for her you need report zero in SS and Medicare wages fields.
Please be aware that according to IRS publication 503 page 7 - http://www.irs.gov/pub/irs-pdf/p503.pdf - You can count work-related payments you make to relatives who are not your dependents, even if they live in your home. However, do not count any amounts you pay to a dependent for whom you (or your spouse if you are married) can claim an exemption.
For your another question that I overlooked - sorry - May you still claim your father as dependent if your mom and daddy file married jointly?
According to IRS publication 501 page 11 - http://www.irs.gov/pub/irs-pdf/p501.pdf - you may not in general claim as a dependent a married person who files joint return - see some exemptions there but seems they are not apply to your situation. Also, if you are living in community property state - 1/2 of wages earned by your mom would belongs to your father.
1. this is correct - however as I stated above if otherwise your mother is your dependent (it doesn't matter if you claim her as a dependent or not) - these expenses are not eligible for Child care expenses deduction. It wouldn't matter if you mother is employee or self-employed.
2. If your father filed a separate return or doesn't file return at all (assuming he is not required) you may claim him as a dependent if other tests are met.
All by All, I think that the difference in claiming your mother as a dependent and claiming Child care expenses deduction would be relatively small. In additional you will need to convince administrators of your dependent care flexible spending account that all above is correct. They may have more strict rules that IRS required. There are other deductions that qualify for Child care expenses deduction - summer camp, some after school activities, etc. Finally, even all these is not a violation of law - it is sounds as not a regular practice and would draw additional attention.