Елена, я отправила твой перевод со своими штрихами на hinative. Тебе, должно быть, будет интересно узнать мнение нейтива.

I saw you speak English so well on Instagram. My friend, teach me English too, for it'll be necessary for my profession. Best regards to your parents! does this sound natural?

This is technically correct, but sounds a little awkward. It is stiff. Here is a more native way of saying it. "I saw your Instagram and you speak English so well. I would love if you would help me with my English, I'm going to need it for my profession." When making a request of someone who we are not close to, we use "softening language". Instead of "teach me English", we say "please help me with my English", or similar. Unless you have met their parents, I would not say anything about them. It is unusual to make this comment in English. If you know their parents, you would word it like, "I hope your parents are doing well".

Странно, Татьяна, а зачем было мой перевод отправлять? ... я понимаю, когда человек свой перевод проверяет...

Елена, я ведь тоже примерно так перевела бы Я подумала, что скажут нейтивы. Извини, если что не так, конечно. Но мы здесь в своём соку варимся.