I would see if you can work something out on your own to get your coop. Whether that's to build it yourself or have someone deliver it, or even offer to pay your parents gas money to pick it up.
If your parents are paying for feed, etc. for your animals, I would be very grateful for that. They might consider that your thing and if your brother doesn't keep pets, perhaps they buy him video games because that is what he enjoys. I don't agree that your dad should hide that form your mother, but I also don't think it is your job to run and tell your mom what he is doing either.
I think what you need to do is sit down and have a CALM talk with your parents. That will show them you are trying to reach out in a mature fashion and surprisingly they may actually listen to you. However, make a list of what you want to discuss and be sure you are not being petty. You can even talk to your father first about hiding things from your mom and express to him that you don't feel comfortable being in the middle of it. Just remember, not everyone is an animal person and that may be the case.