While the music sounds 'hardcore' and yes, the singer does, in fact, have hot pink hair, the lyrics and music aren't angry at all.
The worst lyrics I could find is a quick mention of the word 'whore' in their song "Misery Business" and one of the chorus lines of their song "Fences" that says 'it's obvious that you're dying, dying'. And while, in text, that chorus line sounds dark, in the context of the rest of the song it really isn't.
Most of the lyrics talk about overcoming problems, friendship, and dealing with life. Everything from dealing with love loss to dealing with pressure of everyday life.
Paramore is a great band with lyrics for everyone to enjoy and may actually be a bridge between you and your kids given the band's semi rock-n-roll style while still being something that the average 15/16 year old would enjoy.