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Thank you for your question.
It sounds like what you are trying to do is quite noble. The short answer to your question is -- yes! The family can sign a contract allowing you to spend time with their child outside of school.
The thing you need to keep in mind is that there are some liabilities that cannot be contracted away. The school board's biggest concern was what would happen to their liability in the event you were sued. In the same vein, if you were sued in an individual capacity, you would be responsible to defend the action. Now the parents can agree for the two of you to spend time together and there are some liabilities that you can prevent; however, in the event the child were to an allegation against you, you would have to shell out money to defend against the allegation.
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