It is not through touching their bodies that you achieve a good bond with them (you are thinking dogs, cats, horses) As a matter of fact, people should never touch their birds anywhere else but on their heads, necks and cheeks. The rest of the body is a 'no touch' zone although, most of them (not all, mind you) would have no problem whatsoever with you touching them anywhere if it's bonded to you and trusts you implicitly.
You bond by spending hours and hours and hours with them in their own terms. You let them out of the cage (you don't push them to come out, you just open the door and walk back) and talk, sing, whistle, offer them a treat every now and then and, basically, just keep them company. All parrots are highly social and naturally affectionate so, before you know it, they will be approaching you for closeness and, when this happens, training them is easy. But, for training to be really effective, you need to identify a high value item to use as the reward.