A parent can sign a notarized letter giving you temporary custody of their child without having to go to court. This letter should give you permission to make important decisions about the child's medical care and education. It is suggested that you get a medical release as well in case of any emergencies. This agreement can be nullified by the parents of the child at any time.
In the event of the parents on the child not being available to sign a notarized, private agreement, you can ask to fill out a Caregiver's Authorization Affidavit which will allow you to enroll the child in schools and get authorization to make decisions about the child's medical care.
This affidavit is NOT an official court order but it will allow you to have some rights in the child's welfare If you are related to the child. However this affidavit can be cancelled by the parents of the child at any time.