Given the assault shown on the video (female suspect tackles minor): if NYC (city) does not prosecute, what recourse does the minor's parents to obtain justice? State / Federal?
There's always civil action possible; "assault and battery" has a civil variant as an "intentional tort"
Civil assault and battery are torts. A tort is a wrong committed by one person against another, causing damage. Specifically, civil assault and battery are intentional torts. [...]
Intentional torts are torts that are committed on purpose. In addition to assault and battery, causes of action such as false imprisonment, slander, and fraud typically fall under this category. While the requirements of duty, breach, causation and damages are the same, the added element of the tortfeasor’s intent is taken into consideration as well. If a plaintiff cannot prove that the tort was committed intentionally, it may be a case of negligence as opposed to an intentional tort.
Generally speaking, under US law, the victim or alleged victim of a crime cannot force a prosecution. Police and prosecutors have wide discretion on which crimes to prosecute. Even if it is clear that a crime has been committed, there is usually no right to have police or court action. If officials think that the public interest (or their own interests) would not be well served by pursuing a crime, or that a conviction could not be obtained, they may choose to simply drop the matter.
(An exception: in some states a domestic violence report cannot be simply dropped.)
A request may be made to state officials to intervene, which they can do, usually through the office of the state Attorney General. However, such state-level intervention is unusual.
Federal officials will only get involved if there is reason to think that a Federal crime has been commited. In some cases a state crime can also be a deprivation of civil rights that violates Federal law. When the crime involves more than one state there may be Federal law violations involved. There are various other Federal laws that apply to specific situations, for example robbery of a federally-insured bank is a Federal crime, as is kidnapping. But most crimes do not have Federal aspects.
Publicity and political pressure can sometimes induce local officials to take action on a crime that they initially did not want to address. The local prosecutor's office can often get police response if it chooses to press an issue.