i recently started listening to Avril's three albums: Let Go, Under My Skin, and the recent The Best Damn Thing.
all i can say is:
POP defines her music. there are some tracks like In My World and Too Much to Ask (from Let Go) that is more than good enough to listen to; as well as Keep Holding On (from The Best Damn Thing) which is as powerful as a socially important song can get.
in other words, she's the Britney Spears of Rock. LOL
on the other hand, Paramore leans more on the creative and technical side; still POP and mainstream, mind you, but the focus is not entirely on the vocalist, unlike the former. and probably as the result of their influences, they still incorporate heavy guitars and drums that no doubt labels them as ROCK.
you can't say who's generally better than the other because they're both different. however, you can compare them perhaps in a smaller scale and a specific criteria.