You can for sure break the trust if you keep touching your bird when it obviously doesn't want to be touched. As I mentioned before, best thing to do is to lure the bird with treats. Find out what does it like the most, for example millet, maybe some fruits like apples or grapes, etc... Don't feed it human processed food! Our cockatiel loved cooked rice, that was something we could offer him (we eat very little salt so he could eat it)
First thing I'd like to address is that it doesn't sound normal to me at all that a 2 month old cockatiel cannot fly. Are by any chance their wing feathers clipped (cut)? The bird at this age should be able to fly normally. If their wing feathers are all still there and not clipped, but they still cannot fly, please have your parents make an appointment with an avian vet to address what is the issue and why the bird cannot fly.
It is possible the bird is stressed while outside, especially if it cannot get into the places where it wants to go. Instead of touching (grabbing) are you able to have him step up to you? Keep the physical touches to a minimum at the start, instead just offer him some yummy treats in the beginning. If you absolutely must catch him, rather do it with a towel (closest colour to him, without stripes) so the bird learns that towel=catching, but even this should be as a last resort if you really MUST get him and put him into the cage. Once you do that, make sure you praise him and offer him a treat. You want to disassociate your hands with grabbing. Instead see if the bird steps up to you, if not, see if you can lure him to the hand. Don't force the bird - this is very important. Lure the bird to you, don't chase him. If he's fluttering away from your hand, then stop.
If the bird is terrified of hands, I suggest start feeding him treats, even while the bird is in the cage. Actually the bird should've stayed a few days in the cage, without being touched so the bird learns that their cage is a safe place, it is their home. Never grab or scare the bird while it is in the cage.
As for the chirping and whistling while waddling around - If it sounds like a happy chirping and whistling, don't worry. But take it slow - Having a non tame bird is no easy feat! First we must understand the bird's needs, and that would be security. They are prey animals and we are predators in their eyes. As I mentioned before, they need A LOT of time to accept their new surrounding and to feel secure. Patience, patience, patience. The trust will not be built over night, it's really a bit by bit, and it might take months before the bird starts feeling comfortable and trusting the family.