In Mexico we called them "horas pico" but I think is the same concept because we say "horas pico" if there are many hours or "hora pico" if it's only one hour. I think that will be useful for you to imagine it as the "name" of that hours when traffic is very congested. "Pico" or "punta" will be the name of the hours or hour, so you don't have to use it in singular or plural, It's just its name. Las horas pico en México son las 7 de la mañana y las 6 de la tarde. Es la hora pico, por eso no avanzamos.

Yes, I agree with you that my friends made a mistake. thank you for explaining it. Now I understand. :)

Your friends just have made a common mistake, spanish is so dynamic that it's change with the use that people do with it. So, maybe in the future "horas puntas" will be one exception, but right now the correct ones is "horas punta". In the article you have a link to the orthographic law(RAE); in the case of noun + noun (the second noun is used as an adjective of the first ones) the plural it's make of the first noun and the second remain in singular.