While @pk is very smart, I don't think he addressed the core issues here, and maybe he can follow-up with some of the things I don't know.
To use an FSA for child and dependent care, you must meet the same qualifications as for the Child and Dependent Care tax credit. You may also have to meet additional requirements as an F-1 student. https://www.irs.gov/forms-pubs/about-form-2441
As a side comment, if you are eligible for the Child and Dependent Care tax credit, you should take it instead of the FSA. For 2021, the credit is much more generous than the tax deduction that would result from using an FSA to pay your expenses pre-tax. The rules will change on January 1, 2022, and at that time, the FSA will be a better deal for people over $35,000 income (the credit will still be better at incomes below $35,000).
To claim the credit or use an FSA, you must provide care so that you (and your spouse if married) can work full time or go to school full time. Your student status does not, by itself, bar the credit. Your child must meet the rules for a qualifying child dependent and be listed on your tax return. You must provide the name, address and tax number of the care provider.
What may disallow you is that as an F-1, you are generally taxed as a non-resident alien (unless you have been in the US more than 5 years). Publication 519 says that for a non-resident alien to claim the child and dependent care credit, you must have "effectively connected income." I don't know if your student stipend will qualify. Can @pk comment?
Publication 519 also says that "The amount of your child and dependent care expense that qualifies for the credit in any tax year cannot be more than your earned income from the United States for that tax year." (page 30, https://www.irs.gov/pub/irs-pdf/p519.pdf)
Earned income is compensation from providing a service. Most of the time, graduate students and post-docs are classified as students and they are not providing a service, they are "learning." (Even though you go to work every day, it's not technically considered working, because you are a student.) Now, form 2441 and its instructions allow you to report "earned income" for the purposes of the credit only, for any month that you were a full time student, and this imputed income allows you to claim the credit. I am also aware that some universities classify their postdocs under a category that subjects them to social security and medicare withholding, which results in their income being defined as "earned" under the tax code.
Where I then have a question relates to the statement in publication 519 that you must have earned income. Does this include the imputed income allowed to full time students in the form 2441 instructions, or is it a special rule that non-resident aliens can't use the imputed income rule to qualify. I don't know.
Bottom line is, I think you qualify to use an FSA or the child and dependent tax credit if you are in an employment classification where social security and medicare tax are withheld. You probably qualify even if those taxes are not withheld, using the imputed income rule in the instructions to form 2441. However, you may want to consult a tax professional with experience filing out NRA tax returns.
And also note, Turbotax can't be used to file as a non-resident alien, you need to find another tax software company that offers form 1040-NR.
