Here are the high school summer programs I know of. However, for all of these I think you are only eligible during your last summer before you enter college. They require you to submit your university acceptance letter along with your application:
There are also summer programs for Korean language, if that’s something you’re interested in:
Evidently, there isn’t anything for those who are still in high school and have a few more years until they graduate. Very unfortunate. I sure hope this isn’t your case, but if it is, there’s always your last summer before university. You can start planning it now - the earlier you prepare, the better!
As for safety, that is the reason why I recommended these programs from these prestigious universities. They’re more comprehensive than anything smaller universities will offer to foreign students, you will be placed under the watchful eye of Korean peers and/or program coordinators, and on top of that you will be given the opportunity to reside in their dormitories. My time at Korea University was pretty stress-free because they had their One-Stop Center, a support system for especially foreign students. They were quick with their email replies, and when you drop by their office they’re always readily helpful. They’re super nice too.
It’s very natural for your parents to worry, of course. If you do decide to apply, do it together. Prepare your application under their watch, and once you’re in correspondence with the Korean university make sure your parents know everything you do. Fill them in during every step of the way. For them to have seen the entire process from beginning to end, by the time you’re ready to jet off they’ll be more at ease. And don’t hesitate to email the university for questions you or your parents may have! The more you know, the more secure you all will feel.
As for general advice about living in Korea, I highly suggest getting a phone once you arrive; and when you do, call your parents! For them to know that you will be easily accessible when you’re away will reassure them of your safety. You can either hit up a service to see if you can use your existing phone, but from personal experience many overseas phones are rarely eligible. Try it though, and if doesn’t work out, they have services that cater to foreigners who will be in Korea for a short period of time. Your university will help you out. They will also provide you with a list of important contacts you may need in the worst possible situations, like the police or nearby hospitals.
Overall, you’ll be fine in Seoul; and these universities are located in its most secure districts. Unless you go to Gyeonggi-do, the outskirts of the city, Seoul is safe, robbery and pickpocketing is rare, and anywhere you are you still stay somewhat connected. The entire city is basically a hotspot, but if you’re not registered to a service you may need to rely on wifi, which is not so difficult to come by (though sometimes it is -_-). You will find a pay phone or an ATM in virtually every street corner, as well as convenience stores, but at the very least, make sure you always have your phone (and that you have credit! I’ve been out and about multiple times without my phone, and it can get super inconvenient, especially when you can’t find wifi and when you don’t have your contacts with you, which is generally the case regardless of where you are in the world.)
These programs are full of international students just like you, and the university will provide you with everything you will need to know about living in Korea. This experience will definitely not be a lonely one ;)
I hope this answers your questions, and keep asking if any more come to mind! Good luck!