Perhaps it should, but your daycare provider must be giving you a pay statement based only on the 2020 year. You should request as similar statement for 2019, as well, reflecting your December 2019 payment.
More to the point for tax purposes, you can claim child care expenses that have been pre-paid, but only in the year the child care is provided.
The IRS states in Publication 503, Child and Dependent Care Expenses that
If you pay for services before they are provided, you can count the prepaid expenses only in the year the care is received. Claim the expenses for the later year as if they were actually paid in that later year. [Bolding mine.]
So you’ll be able to claim all three of the payments on your 2020 return, since the care received was in 2020.
