How we cite our quotes:
[Edward to Bella:] "Oh, they already know everything. They'd taken bets yesterday, you know" – he smiled, but his voice was harsh – "on whether I'd bring you back, though why anyone would bet against Alice, I can't imagine. At any rate, we don't have secrets in the family. It's not really feasible, what with my mind reading and Alice seeing the future and all that." (15.57)
Besides being vampires, some of Edward's family members have special skills. As a result of he and Alice's "talents," no one can keep secrets. It's probably a frustrating experience at times, but it likely makes them closer. This stands in stark contrast to Bella's family, since she keeps plenty of secrets from Charlie and Renée.
[Bella to Edward:] "I don't know," I admitted. My dating history gave me few references points to work with. Not that any normal rules of dating applied here. "That's not necessary, you know. I don't expect you to…I mean, you don't have to pretend for me." (15.71)
Edward thinks that Bella should introduce him to her dad, but Bella doesn't want to deal with that potentially awkward scene. This highlights a difference between the Cullen and Swan families. Edward quickly introduces Bella to his family, and he has obviously been talking to them about Bella for a while. Bella, on the other hand, is hesitant to let Edward meet Charlie, and she hasn't even hinted to Charlie that she has a boyfriend.
[Bella:] "Is that what you are?" I suppressed my internal cringing at the thought of Edward and Charlie and the word boyfriend all in the same room at the same time.
[Edward:] "It's a loose interpretation of the word 'boy,' I'll admit." (15.75-76)
What do you think is more awkward: being a vampire and introducing your family to your human girl/boyfriend, or being a human bringing home your vampire girl/boyfriend? They're an odd couple, but Edward's point is that sometimes you have to introduce your boyfriend to your dad.