[@Lucky Knight] Except that they wouldn't be giving her up, nor would they be losing her- they'd be allowing her to pursue her literal birthright, with no guarantee that she'd actually become an Aes Sedai. 

In point of fact, most women who train at the White Tower were turned away after they were trained to the point where they wouldn't kill themselves, if they weren't strong enough, or didn't want to become Aes Sedai. The ones who did become Aes Sedai would always have the option to turn around and go back home, and support their nation/family/whatever. They wouldn't be giving her up- no matter what, they'd be helping her, knowing that she would be able to come back home, the better for it.

Most novices are between the ages of 16 and 20 during this time, meaning that their parents could have the same amount of time with their kid as any noble parent, since most daughters would be married off to strengthen alliances with other nations and cities and the like at that age, especially considering the importance of alliances during wartimes...  so once again, makes 0 sense.