New York has a Hoosick River, sometimes called the Hoosic River and a Hoosick Falls, but also a Hoosic Valley (in NY) where the Hoosick River runs …

That said, the Allegheny River does not run through Allegany County, NY. It only runs through Cattaraugus County before heading back to PA. Allegany State Park is located in Cattaragus County. The Olean Creek, a tributary does go into Allegany County. You never spell it as the Allegany River, even in New York.

Pennsylvania has a Port Allegany, where the Allegheny River runs through. So Allegany isn’t just a New York spelling. Taking US 6 west from Port Allegany will take you into the Allegheny National Forest (PA), which borders the Allegany State Park (NY).