I am curious about what the twins think of their parents in the Empress Amidala 'verse. What are the things they are told to lie to their father about?

The twins have four parents. They recognize this fairly early in life, and are mildly surprised when they eventually realize that other children DON’T recognize this about their own parents. 

The twins’ parents are: Empress Amidala, Darth Vader, Padmé [redacted], and [redacted]. They don’t know Padmé’s last name or Darth Vader’s other one, but they know there ARE other names there. Their absence is obvious. 

(once, one time only, one of them catches their father coming home very late and very injured, stumbling into the apartments with his own blood all over himself and his prosthetic smashed and torn and a dazed, distant expression on his face, and on the other side of the room Empress Amidala drops her datapad and Padmé whispers, “Anakin”) 

They see more of Padmé than they ever do of Anakin (whose name they don’t know, really, honestly, they’ve never heard it once). Amidala has missions for Vader, far-away places to send him for long times, and to the twins, for some time, a father is the kind man cloaked in warm Darkness who drifts in and out of their lives to touch their minds with an overpowering love and show them the proper way to hold a lightsaber and throw a full-grown adult in full armor a hundred yards back with just one push. They always know him in the Force, but he is not often close enough for them to see or speak to, at a certain age. 

And at that certain age, every night that Vader is off-planet Padmé closes the apartment doors and teaches them every secret she knows how to. 

And, very importantly, how to lie. 

The twins don’t always understand why knowing how to lie is so important, and especially not how to lie to their father–about certain thoughts, certain feelings, certain reasons for certain things. Leia thinks it’s just Vader they’re supposed to lie to, because of course the two of them are different men. Luke thinks they’re the same man, just they don’t admit it. Either way, they both know that Mother can be Padmé and Amidala in the same breath, doing the exact same thing for two opposing reasons, but Father is only ever Darth Vader. 

(except when Mother calls him that other name, of course; the one that they don’t know)