No, red carnations have to be received either from another player or 'dad' on Father's Day, and you need red and pink to breed white
I'm still pretty sure you can get red and white carnations by breeding two pink carnations. But I haven't tested it, so I could be mistaken.
Regardless, if you want some carnations of all three colors, I can just give them to you, if you'd like. But if you'd rather breed them all yourself, that's perfectly understandable.