I'm not sure the story quite justifies the Headline - it might just be that parents have more confidence in schools over bullying and therefore worry about it less? Still, it seemed like it could open up an interesting discussion.
The story certainly doesn't seem to justify the headline! But if 50% of parents are worried about bullying, I think that's worryingly high. But since the number concerned about results is only 52%, I think it's a stretch to say parents are "more concerned" about it, since for a survey of 1000 people I suspect the margin of error would be fairly large.
As for the head of the charity spinning this to claim "It feels like our kids have got to do really, really well [at school] to make a success of life. So it’s interesting that’s what surfaces in the minds of parents." - well, being concerned about results doesn't in my mind equate to thinking a child "has got to do really, really well at school"....
Without knowing the actual survey questions, it's difficult to know what the significance of the story is, but I couldn't find a link to the survey on the charity's website.