In the game guide, it mentions that your villagers can gain parenting skills in another way other than going to the happy hut. How else do they gain parenting skill?

I think telling stories gives a little bit of skill...

Happy Hut?!?!?!?!?! That is cute, Harryfan is right, stories do give skill.

My kids run away from the adults when I bring them together for story telling?

I think it's only stry telling.

If the children are running away from the adults and they have 'Listening to story' as their action, their simply running to the ampitheather, which is where the adults tell their story. If their action says 'Running away' then they probably dislike stories.