Hi Shruti,
Moustache Parakeets have the tendency to be flighty and stand offish even when they are hand fed. On top of that, you have two birds together – so they don’t need human companionship – and they have not been treated well by humans. The best thing to do for now is nothing, as far as direct contact or offering something from your hands. They need to be around people, but a quiet corner of the room would be best for right now. You might try covering part of the cage with a sheet or towel so that they have a place to hide. While that goes against my usual advice, these birds need time to adjust and a place where they can see and hear people without feeling directly involved. When you clean the cage and feed them, keep your movements slow and minimal. Approach the cage with your hands by your side or behind your back. Talk softly to them but avoid eye contact if that seems to make them more nervous. Keep all of your interactions this way and with time, they should get calmer. I think this is the best you can expect from this pair. They have each other, but their lives will be better if they can at least stop being so fearful of people. There are some links on this page that may help, too.
Caring For Your Bird
Thank you for asking Lafeber,
Brenda