"I love you! - I know!"
Translation:Ich liebe dich! - Ich weiß!
Would a child say "Ich liebe Sie" to a parent?
Nowadays, no. I don't know anybody who calls their parents Sie now.
100 years ago, perhaps.
It lasted longer in the Netherlands, I think -- in the 1990's, I met a Dutch girl who still called her parents U. But even then, that was becoming rather unusual.