I have to say that it doesn't matter what your city ordinances say, if your parents say no, it means no. You need to respect her decision about it because she is your mother. She probably doesn't want a rooster, they can be very loud, very early in the morning.
Why is it going to be killed? I didn't quite understand that part...Can't you at least try find it a home? You sound like you've given up any hope there. Don't, maybe even post a picture of it here, someone might live near by and can pick it up. You never know.
Your mom deserves to have her decisions respected, after all she's done for you (all mothers).
And I don't think buttering her up by doing chores (that you're already supposed to do, right?) is going to change her mind. I've tried that a time or two. My mom told me that I have chores I'm expected to do, and that I shouldn't do them only when I want something...
I do think it's sad that you got quite attached to the animal and now face the sad prospect of it being killed...but, that's why I suggested you put in some effort to find it a home other than yours if your mom stands firm in her no.