Generally speaking, money is considered to be received when it is actually available and negotiable to you. Because a post-dated check is technically not valid until the date on the check, it would be considered received as of the date on the check. (I believe this principle is valid even if the depositing bank or the paying bank honored the check before the date.) However, if the check was dated in 2022, it would count as a 2022 gift even if you did not deposit it until 2023, because the funds were available to you, even though you did not access them.
In the US, gift tax is not paid by the recipient, (unless the giver owes gift tax and failed to pay it, in which case the IRS may be allowed to recover the gift tax from the recipient.)
The giver may give $16,000 per year without reporting it; more than that must be reported although gift tax is rarely owed. If your parents have given you $16,000 for 2022 and another $16,000 for 2023 by means of a post-dated check, that will probably be counted as a 2023 gift, since technically the check can’t be honored until the date on the check. However, your parents may wish to consult their own tax advisor.
