The old superstition was that a wild bird (not a pet or domesticated bird) flying into your house may bring death. Also I've heard bringing a wild bird's nest into your house may bring death. It is simply not true. I accidentally brought a nest into my house, with a Christmas tree, over 25 years ago, and I did not die. I've several times had wild birds flying in my house -- still didn't die. So the superstition is a crock to begin with.
As far as the dream...the white dove is a symbol of peace and a nice thing to have flying over one's house. When it turns into a parakeet -- I'm assuming by "parakeet" you mean a domesticated budgerigar -- it is turning into a domesticated, friendly, talkative pet. I would put this as a wish-fulfillment type symbolism; your friend dreams of peace coming and instead of just being a "fly over" for the holidays, it is coming to live within her home.
A dream that was meant to express a hope rather than a fear...
but not to my mind a prophetic dream.
The dream symbol books are a load of **** in my view.
If the "parakeet" was another species of the many referred to as "parakeets," get back to me and I'll take another stab at it.
i used to do dream interpretation all the time, i've gotten lazy, but hope it helps a bit