Sure they are. I know because I worked for one last year.
If I tried to apply to a big company for a programming internship, my resume would never even make it to someone who actually has the authority to hire me. HR would trash it.
Startups don't work like that. I've only ever applied to startups small enough that my email was read directly by the founders. So at least I had a chance.
I actually had several offers last summer, though I got all of them because I know a guy who knows lots of startup types.
Now, though, I just cold-emailed a very early-stage startup and they've expressed some interest. Between "I have experience working for a startup" and "I got into Stanford", potential employers who see my resume at least know I'm not totally worthless. But most HR departments would still take me out of the running immediately.
I never said that start-ups weren't hiring high school students. I said that they were less likely to hire than big companies.
Big and medium sized companies have internships for high school students too. They are the ones who can afford to train workers while most start-ups need people who are ready to create a product right away before their funding dries up.
I don't think I've ever heard of a big company having a programming internship for high school students. They have internships in other fields, but to my knowledge, programming ones are rare to nonexistent.
Correct me if I was just ignorant to them, though.
Obviously the number of these internships is limited. But that is also my point: a high school student (not you necessarily) is rarely suitable for creating commercial products until many months or even years of training. That is why people hire college graduates.
I don't think a startup is likely to give you an internship (perhaps even less) than one of the big companies.