Cecilia is the daughter of the one and only Pansy Parkinson, brought up without a father. Her mother went a bit wild after Hogwarts and Cecilia is the result. Lia has no idea who her father is, but is curious to find him once she leaves Hogwarts, she just hopes he isn't a muggle, not that she hates muggles, just that would be very awkward. So she was brought up alone and her and Pansy are very close, more sisters than mother/daughter.
Cecilia is fiercely independent and has a short fuse when she thinks someone is being a bit of a wimp, though her mother has taught her that if you speak your mind in dramatic situations people might not like you very much.