That is very frustrating and we are glad you reached out to get some help with this.
In the end you may not be able to convince your parents no matter what, but give this a try. Instead of just asking if you can stay home alone ask how you could earn the privilege to stay home alone. By wording it this way, you can demonstrate that you are willing to work to prove to them you can do this.
You could earn trust, show good work ethic, and perhaps be able to stay home alone. You have already done a lot with the extra housework, see if there is more you can do! But this is by no means guaranteed. They still may say no, which is a bit of a downer.
If they simply continue to refuse to let you stay home alone, then you will have to obey their wishes. If you don't then you will only damage the level that they trust you even further. The good news is that you are nearly at an age where you can get a job, move out of the house, and begin to lead your own life.
If they have you stay at the relatives place, try to stay busy by volunteering at an animal shelter, getting a part time job, doing extra curricular activities or spending time with friends. That way you can be out of their messy house as much as possible, making the time you are there less stressful. Also try to focus on your own thing when you are at their place. Write a story, read a book, do anything to escape from the reality of the situation so it goes faster.
We hope that you can find a way to make the time productive and more pleasant. Good luck with your parents!