Yeah, parents can make their kids stress, even if it's not intended.
I remember very clearly the first time I felt stress. I had a stomach ache and a competition 2 or 3 days later. When I told my dad about it, he asked me if it was because I was stressed. In my head, at that time, I wasn't stressed. I just had a stomach ache. Now, when I think about that one little sentence, I think I perceived at that time it as if I was supposed to be stressed...
Since then, I was a big mess in the week leading to a competition, and was never really good in coping with stress. I was having trouble sleeping, concentrating in school and if I had a bad training... watch out. I learned to deal with it with time. Now the stress is different, I've found what's helping me. I visualize, take baths, exercise... Your gymnast should be able to do the same with time.
Maybe you could talk with your co-coach about the emphasis he/she puts on winning.