An MLB player can be placed on the Paternity List when the player leaves his club to attend the birth of his child. (There is no Paternity Leave for minor league players).
A player must remain on the Paternity List for at least 24 hours, but no more than 72 hours.
The Paternity List functions like the MLB 15-day DL or the Bereavement List, in that a player on Paternity Leave does not count against his club's Active List, so he can be replaced on the 25-man roster while on the Paternity List, but he does count against his club's Reserve List (40-man roster), and he does continue to accrue MLB service time while on the list.