Well I did think that maybe their parents were rebels and one tried to flee District 2(which as we know is loyal to the Capitol) to District 13? Maybe that parent ends up captured after somehow making it only as far as District 10? Of course the Capitol would have either killed that parent or made them into an Avox, but I suppose maybe they kept the child alive with somebody else since every child to them is potential tribute material. I figure this theory works since I remember in one of the books there were a few District 8 citizens who tried to flee their district. 

As for my character, I figure this would be more effective if the other parent who remained in District 2 was also killed or made an Avox due to the Capitol believing they helped their partner escape. That in effect then makes both children having been adopted by other people as babies, thus knowing nothing about their past or their real parents.