I have a Frozen theory. I know they're all over the Internet, but I can't hold it back anymore.

So, here goes:

The father must have ice powers also. He learned to control them eventually, but then when  Anna got hurt he panicked and convinced himself that keeping Elsa locked up was the best idea. Notice that the troll never said that the girls couldn't play - he actually specifically said that he "leaves the fun." The father is the one who put the gloves on Elsa and taught her to "conceal it, don't feel it" (the mother just stood there). His portrait hangs in Elsa's room, and it's that portrait that she looks at in "The First Time In Forever." She is trying to live by what he taught her, and stresses over being the perfect girl he wanted her to be. That's why she says "No right, no wrong, no rules for me - I'm free" and similar lines throughout the movie.