If I have parents abroad (South Korea), can my children go to South Korea for free through an invitation letter?

You can go, but only in the name of visiting relatives or traveling. Because you are an adult and have a family, you need to provide certain proof of funds before you can go. It is not free! ! ! The reason is as before: the Korean government has reason to suspect that you have the intention to immigrate!

For the invitation letter, you can go over and provide a corresponding amount of deposit according to the type of invitation letter to prove that you can pay all the expenses during your stay in Korea.

Or you can ask your mother to vouch for you because She has Korean nationality and can be a guarantor. South Korea implements an anti-signing system and requires a guarantor. Students are generally sponsored by the school, but it is difficult to sign if you are an adult and have a family!