Hello and thank you for your question.
I am a different expert and I would like to clear up a few things.
1) Wages you pay to a parent will not be subject to social security or Medicare taxes under your circumstances. This is because, before you include wages paid to a parent as a household employee are subject to social sercurity and Medicare taxes, you must meet two qualifications. The first you meet, which is that your mother would be caring for a child under the age of 18. The second you do not meet, which is that you must be divorced or unmarried, a widower, or your spouse must be disabled (I assume your spouse is not disabled). See Page 5 of the following Publication for more information: https://www.irs.gov/pub/irs-pdf/p926.pdf
2) Yes, you could pay your mother enough to make it so she is not your dependent and qualify to purchase a policy on her own through the Marketplace assuming she meets any other criteria aside from income for doing so. Your mother is probably eligible for Medicaid at this point though too, so look into that before you go purchasing insurance through the Marketplace: http://obamacarefacts.com/obamacare-immigrants/
"For Medicaid and CHIP, certain “lawfully present” aliens no longer face key welfare reform provisions, such as sufficient immigrant sponsor income or a five-year wait."
3) I assume the best method for you is probably to get Medicaid, but assuming she really can't get it and you pay her enough to go through the exchange, then you should note the following:
a) You may generally use your FSA account and/or claim a credit for child and dependent care expenses under this scenario (IRC 21), noting your parent is not a disqualified related person per the example in Reg Sec. 1.21-4(c):
Example 1. During 2007, X pays $5,000 to her mother for the care of X's 5-year old child who is a qualifying individual. The expenses otherwise qualify as employment-related expenses. X's mother is not her dependent. X may take into account under section 21 the amounts paid to her mother for the care of X's child.
b) You would lose the ability to claim your mother as a dependent (IRC 152(d)), which would generally cost you about $1,000 in taxes, though it's impossible to say without all the specifics.
c) You have no federal employment tax filing obligation unless you withhold federal income tax from your mother's pay. If you do withhold income tax, which I don't suggest, then file Schedule H with your Form 1040.
d) You would not supply your mother with a Form W2 at year end (Per IRS Pub 926, p 10), unless you withhold federal income taxes.
e) Your mom must file Form 1040 and report on line 7 the wages that she received from you despite having no W2 (so be prepared for your tax software to get "fussy" on you).
f) For your mother to get a premium tax credit as you desire, she must file a joint return (IRC 36B(c)(1)(C)), but you cannot claim your father as a dependent if he files jointly with your mother (IRC 152(b)(2)), so be prepared to lose that exemption too (which will cost you roughly another $1,000 in tax). Note - There is an exception that allows you to claim as a dependent a married taxpayer filing a joint return solely as a claim for refund, but the Code only acknowledges this for qualifying children (IRC 152(c)(1)(E)), not parents, even if the IRS may acknowledge it for parents too as an administrative election (see pub link below - joint return test), so try to claim your father as a dependent after he files a joint return with your mother at your own risk. It's unclear how this might be interpreted at this point and the IRS may get tripped up over it. https://www.irs.gov/publications/p17/ch03.html#en_US_2014_publink1000170864
g) Register for and pay any applicable State employment taxes (see both links, one is for withholding and the other unemployment):
I hope this information is helpful and clears things up. A positive rating is appreciated (time spent to answer your questions with proper citations = about 1.25 hrs, my normal hourly rate = $150, so Happy Holidays!). I hope you find this fair, but for further clarification, please be prepared to accept an additional services request at my normal hourly rate given the amount of time I've put into this already. Again, thank you for your question.