Can your landlord discuss your lease with your parents?

Yes. A lease is not protected or confidential in the way that medical or other information would be. A landlord is free to discuss your lease with anyone they want, unless your contract provides otherwise.

However, as a practical matter, most landlords, and most business people in general, opt not to discuss the terms and conditions of a contract with other parties.

Additional: If you are a minor and your parents co-signed, or are in some way responsible for you and/or the rent payment, they can speak with them.