Have your doctor write down the diet he has specified. It sounds like your parents think you don't like the food they are preparing, and are being difficult or rude. If that is not the case, then sit down with them and explain the details at a time other than dinner time.
Some people think a meal is important and the time put into it is important to appreciate. Hopefully, you can show appreciation, yet still eat what is beneficial for you. Tell your parents how much you appreciate these efforts and everything they do for you, but need to eat as the doctor has "prescribed". This is like a dietary prescription, and is important.
Your parents also may feel a little threatened that the doctor could be suggesting their foods are not healthy. That may also play into their reluctance to allow you to eat as suggested by the doctor.
Another option could be to eat the foods recommended, along with just a small portion of the food prepared by your family. Usually, if you eat a good quantity of the healthy foods, they will help even if you consume just a bit of the other food. Just don't indulge, too much.
Speak to your doctor about this.