First of all, I feel like HN is going to give you a skewed perspective here. As far as I've seen, the HN community is way more GOGOGO DO A STARTUP than most, and the skill sets and life paths they value might not be the ones your parents value.
Secondly, I've done both. I've started my own company, which got rejected from YC and ended up in another incubator, DreamIt Ventures, and ended up crashing and burning due to a variety of reasons. I've also spent a summer, and now the better part of a year working at a YC company, Amicus (totally check it out, its an amazing place to work.) I'd say I was incredibly happy with both experiences, but I gained different things from each one.
YC, or the accelerator experience, puts you in the hands of some top start-up gurus and mentors, creates sky high stakes, and gives you the opportunity to build something really awesome yourself. If you have previous startup experience, an idea truly worth the immense work, and/or a co-founder whom you trust with your life, this might be for you. When asked if I'd do DreamIt ventures again, I always say yes, but I warn people that its way harder than you first expect, and if the mix is a off with your co-founder, it can be a few months of extreme pain and constant difficulty. You learn way more about the ins and outs of running a company, and the feeling of having your work criticized to no end, and of failing despite everything you tried. This is a good way to improve certain skills, but I'd argue not all skills fall under this category.
The intern experience is more flexible. From everyone I've talked to, it seems that most companies are looking for especially driven young people who will try really hard before giving up and who will generally do more than their employer asks. This means that once you have the spot, your employer probably isn't going to put much stress on you to work incredibly hard, and while the culture might demand that you work more than 9-5 every day, you generally can take weekends off, or take a day off here and there. This means that if your goal is to learn how to run a software team, as well as pick up a bunch of different tech in your free time, this is for you. If there's a hackathon in the city you're in, you can part-take, and if there is a tech talk that will take an extra day, you can almost definitely get out of work to go check it out. One of my friends at Amicus spent every weekend hacking on a side project, and had an incredible portfolio by the end of the summer, as well as new knowledge of how to write clean code and manage an engineering team of more than one or two.
Now its not to say that you can slack off at work, or not take one of these internships seriously, but at the end of the day you're either trying to 1. Learn a lot or 2. Get a permanent position at the company you're interning at. For the first goal, theres a lot of flexibility to put down your internship work when you go home, and build that cool project you haven't had time for, and for the second, a quality employer will judge you by the quality of your work, not your hours.
Thus, you really have to think about what you want to get out of an internship. Do you want to learn from the engineering pros, get feedback on your work, and build side projects, or do you want to learn to pitch as well as push code, do UX research as well as debug, and have a company in your name instead of your name on a company. Both scenarios are amazing opportunities, you just have to figure out which one fits you better.