I had a slightly similar situation - here it is. My husband and I had custody of his very young son whose mother had run off. She floated in and out of the picture, but even in her absence it was very hard to exert my own authority with my 'son.' He had a mom, and I wasn't her. He had paternal grandparents who thought they were his parents and muddled the lines of authority, as well. In some ways I was the nanny in terms of how my authority worked. I think the tactics I used with success might work for you, as well.
Tag team the child. This does not imply aggression. It means identify your team and get them working on the same play. Your team consists of:
1) The child (who is benching herself, count her out for the moment).
2) The parents
3) The teacher
You need all of these players to work in unison using the same game strategy to effect the same goal: Getting this child to work to potential.
Talk to the parents and get them to set up a meeting with the teacher. That's their job, not yours. Together, the four of you need to brainstorm a few strategies - to be tried one at a time. All of you need to be comfortable with the strategies agreed upon; all of you need to commit to unwavering CONSISTENCY and team support. You need to agree upon weekly or daily emails to each other, monitoring progress. Seriously, keep each other abreast of progress or the child will rediscover her leverage by playing one against another.
I guarantee that the little girl will quickly grasp the fact that she can no longer play the parts against the whole. She will come to understand through your loving but firm team actions that the jig is up. Time to get cracking. Play time, darling child, is for after the work is finished.
I found that if I steered clear of threatening tones or language, my son understood that 'all us grown ups' acting in concert for his betterment was a true expression of love. I made sure I mentioned it point-blank every now and then, too. "We do this because we love you."
I'd also suggest that you make it clear to the parents that making a team effort like this is the ONLY option. It is not possible (been there done that) for you to do this without their full, coordinated cooperation.
Look, this child has three environments in which she lives: school, your house, and her own home. All three environments need to be delivering the same message in the same language with consistency. Accomplish that, and you've got it - and the homework -nailed.
Best of luck. :-)