If a family is travelling with infants under 2 years old, these infants will not need a seat (of course you can buy them a seat if you want), and they can just sit in their parents' laps. Given that, why do airlines prohibit two infants from sitting in the same row (for example Seat A,B and C) at one side of the aisle? Why is it that one of the parent has to move to another row or to the other side of the row?
This is prohibited for safety reasons. Each row has one extra oxygen mask (there are exceptions, some rows have no extra masks and they will be marked somehow). If more than one infant is seated in a parent's lap and the oxygen masks were to be used one of the row occupants will have no oxygen mask to use.
Usually the reservation system/agents will avoid such a situation but it could happen that a family with two infants will be given boarding passes in one row, the cabin crew then will move one of the parents along with one of the infants to the other side of the row or to another row.
Update: in some larger jets the set of seats in the middle (4 or 5 seats) may have two extra oxygen masks, in that case two infants can accompany their parents in their laps.