Which sentence is correct?
My parents don't like my playing loud music
My parents don't like me playing loud music.

I have heard it is correct with my, but if it is, why?

They are tricky, but simple once you change a word or two to put them in different contexts.

On the first one, yes I do see why it could be correct.
The word "MY" is a possessive adjective. A noun comes after the possessive.
My house.
My car.
My Answer.

Probably the word that is confusing you is the -ing (PLAYING). most likely you are thinking this word is a verb when in reality it is not working as a verb. we have here a noun. the complete noun is "Playing loud music". This phrase is a participle phrase functioning as a noun. Try this on for size.
My parents don't like my room.
My parents don't like my stereo.
My parents don't like my music.
My parents don't like my playing loud music.

The second sentence talks about you doing something and your parents not being happy about it. Another way of saying it would be:
My parents don't like that I play loud music.
Once again, from the word "that" to the period, we are talking about a noun clause. Clause is another word for sentence.

Hope it helps

My parents don't like my playing OF loud music. OF is th operative article, my is correct.

My parents don't like me playing loud music. if you were to go for the first option then you would need 'of' after 'playing', this is because you are possessing something with the use of 'my' but not expressing what you are possessing. the first one would be acceptable in a spoken context.

I've never heard of it like that. I think "me" is the correct way.