I do think you will need to talk with your parents and be truthful about everything, anxiety, depression, severe culture shock, etc. They are going to worry, there’s no way around that, but they are also some of the best people to help you. Keeping it from them isn’t working. While a certain amount of discomfort is expected in your situation, discomfort that leads to growth, your level of discomfort is affecting your functioning in a way that is concerning.
In the meantime you have to figure out how to get adequate nutrition. Not eating well will only aggravate your symptoms. Many Japanese foods are an acquired taste of they are new to you, especially a lot of the fishy things often cooked at home. Since you are in a city, have you tried searching out some noodle shops? Often small places, inexpensive too. Ramen in particular is easy to find a flavor that you might enjoy. Udon as well. Also, Seven Eleven, Family Mart, or Lawson Station, all convenience stores, usually have great food options at low prices. You could even offer to cook an “American” meal for your host family every once in while if you have access to a grocery store.
Try calling the telephone help line at Tell Counseling linked above. If you are in Tokyo, you may even be able to get in to see someone there.
Keep reframing your thinking in the positive. Remember why you wanted to do this in the first place. Work on the language experience that you went there to get.