Generally speaking, a parent has little ability to sue a court appointed attorney for the children for two reasons.  First, there is a lack of privity and second, court permission is often required before a court appointed attorney is subject to legal malpractice lawsuit.

Swerdloff v Swerdloff  2022 NY Slip Op 01279 Decided on March 01, 2022
Appellate Division, First Department is an opaque decision which seems to set the bar on whether a parent is permitted to sue at whether the Court specifically prohibits the suit.

“Order, Supreme Court, New York County (Michael L. Katz, J.), entered December 7, 2020, which to the extent appealed from as limited by the briefs, granted the cross motion of the attorney for the children, Dawn M. Cardi, Esq. (the AFC), for an order directing defendant mother to pay her pro rata share of the AFC’s outstanding invoices by December 28, 2020, unanimously affirmed, with costs.

The court properly directed both parties to pay their pro rata shares of any outstanding invoices to the attorney representing the children in these divorce proceedings (see Matter of Donna Marie C. v Kuni C., 134 AD3d 430, 431-432 [1st Dept 2015]). Further, the order did not expressly or otherwise prohibit the mother from challenging the amount of the AFC’s invoiced fees or asserting a defense of legal malpractice, and therefore was not improper on the basis that it deprived the mother of those rights (see generally Venecia V. v August V., 113 AD3d 122 [1st Dept 2013]).”