I think we get so strongly attached to our pets because we have so much trouble attaching to people. My dogs are my life. My old dog died in August and I was completely useless for a couple of days and I still can't talk about it without crying, like now! I gave her all the loving and good care and attention I should have gotten as a child but didnt' get. Her box of ashes is still sitting on the mantel and I just can't let go of her yet even though I still have the other dog and already got a second. He's doing his level best to brighten up my spirits, and those of the other dog who is also still depressed over losing the old one.
It's good you got another bird, one that doesnt' look like the first one. Each pet deserves to be loved for itself and not just used as a replacement for the one that died.
My mother did the same thing as your parents, not allowing us to get attached to pets without getting rid of them. It's like she didn't want any competition for our love even though she never ever got my love, having none to give to me herself. All I had in that family was the dogs and they always were taken from me. Must be a common characteristic of ACOAs. kids of alkies raised with no love from parents turn all their love to their pets.