I know that i can't really bond with two parakeets, however i would like them to at least not be afraid of me. every time i step near them or put my hand in the cage they start rapidly flying around or breathing hard. I am such an animal lover so trust me I'm not doing anything to hurt or scare them. When they bite me because they are afraid, I listen, and when they fly away from my hand i keep it there for a few seconds to let them calm down then slowly bring my finger back up to them while softly talking. I've only had them for about 6 months. Is it possible that they just need more time to adjust to me?
You need to me in the room near them for an hour or two for a few days or until they are calm. Then you can just open and close the door for a few days until they are ok with you doing that. Then, and only then, would I recommend you put your hand into the cage. Just put your hand on the floor of the cage for a while and gently talk to them, do this I till they are calm. You can start moving your hand around or closer to them now, and continue this till they are calm. The next step would be to teach them to step-up on your hand and for that just hold your hand with a treat like millet on it and wait for them to get on your hand to eat it, they will hop of when they are ready. Hope that helps! You can find much more information online by searching "taming parakeets".