The pension should not be affected by any outside income.
If she is less than her full retirement age (65-67, depending on when she was born), then outside income more than $17,000 (not counting the pension) will reduce her social security benefit. If she is full retirement age or older, outside income won't reduce her benefit.
Outside income (pension plus other work) can make the social security taxable. See this. https://ttlc.intuit.com/questions/1899144-is-my-social-security-income-taxable
Then, you need to decide if she will be your employee (provide care in your home on your schedule as your household employee) or an independent contractor (provide care in her home, certain other factors influence this test.)
If she is your household employee, you pay wages and give her a W-2. You generally do not have to withhold social security and medicare tax (household employee tax) from a parent's wages, but see the links below. She would file a tax return and add the income to her pension and SS income to determine her total amount of income tax owed.
If she is an independent contractor, you give her her pay and she reports it as self-employment income. She pays 15% self employment tax, and may or may not pay income tax depending on her other income. She can also reduce her net taxable profit by expenses such as food, and part of her rent (if she provides care in her home) which will reduce both her income tax and self employment tax.
You have to use her SSN on your tax return to use the FSA benefit, so the IRS will know to look for matching income on her return.
You can use up to $5000 in FSA benefits for one child. If you paid your mother $5000, then the FSA will save you around 25% ($1250) and probably cost her about the same in income tax, unless her pension is less than $6000 per year. So overall for your family, it's kind of a wash. If you pay more than $5000, then the amount over is just money out of your pocket and taxable income to her so you both lose. (The only reason to pay her more than $5000 is if she wants it for her time and trouble, in which case you may want to report $5000 in income and treat the rest as a "gift".) To figure the exact tax impact, someone would have to test her tax facts with and without the income.
https://www.irs.gov/taxtopics/tc756
https://www.irs.gov/publications/p926#en_US_2017_publink100086735
