I believe the parents being upset about their daughter having the mana illness has more to do with them finding it embarrassing to have a child with no mana rather than that its a hereditary disease. That's why Lance got so upset, it's clear the parents only care about their children to the extent that it makes them look good. The child with two lines is favored and the child with one line that is quickly vanishing isn't worth looking after as it's an embarrassment to their family.