First of all, the money may already be lost. Unless your employer plan already allows for a grace period or carry forward, the money is gone. If your plan allows for a carryforward of the unused balance, you have until the end of the carryforward period to spend the money. You may want to check with the plan administrator first to see if care paid for in January or February 2021 is eligible for reimbursement. (If it is, it will be reported on your 2021 tax return next season, not this year.)
Then, care is only a qualified expense if provided so that you, and your spouse if married, can work or look for work.
You are allowed to pay your parent (the child's grandparent) for care and get reimbursement. You will need to show some documentation to the plan administrator, probably including a W-9 form reporting her social security number. Since she will be living in your home, she is your household employee, but you are not required to issue a W-2 as long as the wages are less than $2,200, and you are not required to withhold or pay household employee's tax since she is your parent. Lacking any tax reporting paperwork, your mother would be on the honor system to report this income and pay tax on it. If this is her only income it would not be taxable even if reported. (As a green card holder she is required to file a US tax return and report and pay tax on all her world-wide income, I'm surprised she has no other income anywhere. Maybe stays with other relatives. Not really a part of this question.)
