If it's the type of parents, how would suggesting an alternative name to them change anything?
Personally, I don't mind that
Lily's really popular. It's sweet without being too sweet (I think the L's give it a certain awkwardness), and some names have to be more popular than others.
Lily's nice. I am glad I have managed to overcome the "images" associated with names when I am considering whether they are good names or not; it enables me to be happy to meet a
Lily,
Emily,
Emma,
Victoria,
James,
Jacob, or
William.