You would do better with a goat. Chickens will eat selectively, leaving behind a lot of the greens that they don't like. Over time, this will result in a very patchy lawn, made up mostly of noxious weeds.
While your argument to support getting chickens and rabbits is commendable, your parents will most likely see it for what it is: a creative way to get the "pets" you would like to have. You would be more successful if you convinced your parents that you possess the maturity to take care of a small flock of chickens. This includes not only providing for their daily needs, but earning the money to buy their feed, and being able to build their housing. This is not outside the capabilities of most teen agers.
I started my first flock when I was 12 years old. My dad was instrumental in this. But, within several years, I was earning all of my personal money, even buying or making most of my clothing.
Does your family keep a garden? If so, chicken and rabbit manure make great garden amendments. Eggs can also be sold to cover the costs of the chicken feed. You could also sell chicks, and possibly baby rabbits depending on the market in your area. If you do not have a family garden, perhaps you could put in a couple of small garden beds, and sell extra produce to cover the cost of your animals.
Do you have the time to undertake the work involved?