Technically, military rockets were not a thing that existed at the time of the US war for independence. The phrase "the rocket's red glare" refers to Congreve rockets, which were invented in the 19th century following the introduction of rockets into modern military use in India at the end of the 18th century.
The Congreve rockets were inspired by the rockets used by Mysore starting in 1780. So there were military rockets in use at the time of the US war of independence, they just weren't being used by either of the combatants in that war.
https://en.wikipedia.org/wiki/Congreve_rocket