I've actually never towelled my green cheek conure to clip his nails. When I first got him, I started training him to lay on his back and let me clip his nails. I flip him on his back with one hand, and clip a couple nails with the other. He gets a treat and a brief break after every nail clipped. He's not stressed out by nail clipping because he always gets special treats. This takes a little while... nail clipping generally takes a couple days for me, since I clip them just a little bit at first to get the quick to recede, then clip a little bit more a week or so later. This whole training process did take weeks, though... I wanted to be careful about making nail clipping a positive experience. I trained a little bit with him every day before I ever actually managed to clip a nail. Now, every time I put him on his back, he sticks his toes up thinking it's clipping time, lol.
Maybe you could try something similar with your Amazon, although it takes a while. Maybe you could train her to let you put her on her back in your lap. Then you could start to reward her for letting you put the nail clippers near her toes. Slowly move up to clipping the tip of one nail then giving her a reward. She may also prefer nail files, but you'll have to train her to slowly tolerate longer and longer periods of filing.
My main problem is that my conure likes to try to play with the nail clipper while I'm trimming his nails... little pain in the butt! lol.