Generally, with an F1 visa, you are considered a non-resident of the US for the first five years of the F1 visa. You would file a nonresident tax return form 1040-NR as married filing separately. TurboTax does not support this form, you will need to use another software or find a professional tax advisor. Your spouse would file form 1040 as married filing separately, and if they qualify to be their parent's dependent, they would check the box that says "I can be a claimed as a dependent by another taxpayer".
You and your spouse have the option of choosing to treat you as a US resident and filing a joint return. This means that you must pay US tax on all of your worldwide income, including income from other countries that would not be taxed on a nonresident return. It also means that you would qualify for certain tax benefits and a lower tax rate for filing a joint return.
However, this may impact the parent's tax return. If you and your spouse file a joint return, then your spouse cannot be a tax dependent of their mother. This does not change the parent's ability to help with education expenses, of course, but it would prevent the parent from getting tax benefits for paying for the child's education. If your spouse were to file a joint return and not be the parent's tax dependent, it might actually improve your spouse's ability to qualify for financial aid. But it would definitely impact of the parent's tax return to no longer be able to claim their child as a dependent.
You may want to consult a tax advisor together with your spouse's parent to determine what is the best overall strategy for all of you.
