My advice to you would be to bide your time and be patient. Do the very best you can with what you have at the time. Take very good care of your chickens and give them as much of your time and attention as you can.
I have been a kid, and I have had kids, so I do see your parents point of view as well as yours. Show them that you are mature and responsible and that your chickens will be well-cared for and neat and clean. No one wants a mess in their yard and no one wants to end up looking after their kids’ abandoned pets.
( I can remember an incident with my one son and his ferret that he swore he would take care of, and guess what happened.)
When your parents see that you are seriously devoted to your chickens, and caring for them, even after the “novelty” has worn off, they might be inclined to consider your thoughts on improving your coop and run.
Another idea which would be cheaper and less obtrusive. Buy a roll of cheap fencing, like chicken wire, that you can unroll into a circle to make a play yard. That way, you can let the chickens out to play when you have time to watch them. When they go back in the coop, you can roll it up again And put it away.
i did this for my ducks when they were smaller so they could swim and play. I’ll post a photo Or two. You might have to enlarge the pic to see the wire.