That's not the case in California:
[SUP]11[/SUP] California Welfare & Institutions Code Section 903.1 provides: "(a) The father, mother, spouse, or other person liable for the support of a minor, the estate of that person, and the estate of the minor, shall be liable for the cost to the county or the court, whichever entity incurred the expenses, of legal services rendered to the minor by an attorney pursuant to an order of the juvenile court. The father, mother, spouse, or other person liable for the support of a minor and the estate of that person shall also be liable for any cost to the county or the court of legal services rendered directly to the father, mother, or spouse, of the minor or any other person liable for the support of the minor, in a dependency proceeding by an attorney appointed pursuant to an order of the juvenile court. The liability of those persons (in this article called relatives) and estates shall be a joint and several liability. (b) Notwithstanding subdivision (a), the father, mother, spouse, or other person liable for the support of the minor, the estate of that person, or the estate of the minor, shall not be liable for the costs of any of the legal services provided to any person described in this section if a petition to declare the minor a dependent child of the court pursuant to Section 300 is dismissed at or before the jurisdictional hearing. (c) Fees received pursuant to this section shall be transmitted to the Administrative Office of the Courts in the same manner as prescribed in Section 68085.1 of the Government Code. The Administrative Office of the Courts shall deposit the fees received pursuant to this section into the Trial Court Trust Fund."
http://www.shouselaw.com/rights-parents-minors.html
In many states, if the parents have the ability to pay, they are responsible for their child's legal expenses.