They thought the school were partly to blame.

The Bakers thought that the school had a part to play in the suicide of their daughter for these reasons.

1. They thought that the school did nothing to prevent bullying. When Hannah's mum goes into the school toilets and sees the writing on the wall calling various girls "Sluts" and "Cum Dumpsters" she sees this as evidence of an environment where bullying takes place, and by leaving the writing on the wall the school are aware but not taking any action against it. The list of Best and Worst that Alex created is also seen as evidence of an environment where people are shamed for physical appearance.
2. They thought that the members of staff should have noticed that Hannah in particular was not in a good frame of mind. Her poem that was published in the unofficial school zine "Lost and Found", was recognised as dark and a call for help, however no one acted upon it.

3. The school had a counsellor that as part of his job should have been to pick up on signs that something like this might happen with a student. They will probably feel even stronger that Mr Porter should have acted differently when they listen to tape 13.

So in summary the school allowed an environment to grow where bullying, shaming and lying was seen as acceptable and if a student needed help they failed to recognise or act even when the student tells a member of staff

that they were raped and want to end their life.

Here is a website that talks about reasons a school could be culpable and sued:

It looks that the parents would be trying to sue the school district under Failed Duty.

Failed duty. The school usually isn’t legally responsible if a student assaults a classmate out of the blue—without a previous history of harassment or misbehavior. In order to prove that the school is liable for failing to protect a child from peer harassment, one usually needs to show that school officials:

• knew a particular student or group of students had been bullying the victim
• could have predicted that the bullies would do it again, and
• didn’t take reasonable steps to prevent the damage that resulted when the bullies repeated or escalated the harassment.

To start with they were probably clutching at straws a bit. But as details of the school environment became known their case would have grown stronger. With the tapes coming into the hands of the Bakers they then have very strong evidence to show the school did fail in its duty on a number of occasions.

This suing schools for bullying happens in the real world a lot:

Teenage boy sues school

Jury Verdicts and Settlements in Bullying Cases

After Her Son Was Bullied, Mom Sues the School District