Bleh.
Im not sure gift is appropriate in either case.
By the OP logic, neither is the gift. The baby is the gift from God.
Using further God logic, one would assume that if God, all powerful, wanted to give the gift of a child to someone, then his aim would pretty much be on target. So if the baby were a gift to someone else.. He probably would have given him or her to that particular person. But he didnt.
SO... once the baby is here... the person, having the gift of freewill, can choose to accept or deny the gift that he has given.
If anything... the baby is a re-gift by this logic.
And Im sure we all agree that the baby is not a re-gift.
And the family is not any kind of a gift as they are simply looking to fulfill their desire to be parents... so they are looking for the people who are denying the gift that God gave them in hopes of being the recipiant of the regift... as chosen by the original recipiant and having no affiliation with God whatsoever.
SO..
God=Gift Giver
E-parents=recipiant & possible regifter.
A-parents=recipiant of re-gift. Not the person originally intended to have the gift.
Please note: this is not my view
(as you all know...) it is just the logic layed out here.
Hopefully now, no one thinks that anyone is a gift.