I suggest you show mom & dad this site and let them read our collective responses and other threads here in their own time, we don't know you from a bar or soap wrt to how mature and responsible you are , your mom and dad do. At the very least they'll learn that lizards are more than just dump animals governed only by instinct and reflexes to stimuli, that they are more like cats and dogs when it comes to their intelligence and ability to learn. <-- I told my father that they are very intelligent and since we can't get a decent dog, we could get a lizard as a compromise of some sorts.
Pestering them about it is a sign of immaturity .... but that's my opinion. <-- I never bother my parents about something for longer than necessary. Usually I'd just sit them down and tell them why I need something and promise to pay for it in my own money.
Caring for even one bearded dragon (even an adult) is very time consuming , you are not looking at buying a gold fish here, but a sensitive emotional being who will rely on you for everything - EVERYTHING - it needs for it to live and it will need lots of interaction too (it's very unfair to the beardie to keep it forever inside a box and never allow it to socialize with you or your family, it's a huge responsibility . Horses depend on me too. There's four of these 1300 lb babies there, which I take good care of. They are always fed and happy.
In fact, one of the gazillion reasons I didn't want to get a fish (even though some of them are absolutely gorgeous) was because they have the memory of my Nutella jar, thus they won't remember me, love me, or even downright care. And yet they do need to be cared for themselves. I, for one, had always wanted a pet that would understand that I care for it, allow me to take it out of the box/tank/outer space and socialize with it. About the family part - my siblings would love to interact with a living, breathing lizard that looks like a dinosaur - under close supervision, obviously. It would also be a great opportunity for them to watch and learn about how living things need to be cared for. They will also have to understand that even though some creatures are very social, they still need their personal space. As for my parents, I think they'll get used to Torty the Second Just Different being in the house and eventually start interacting with him too.
It's very common for children to loose interest in , become bored with their novel pet, and more often than not it's mom or dad who have to take over caring for the pet , and if it's a lizard, they often simply release it, this is a death sentence to a tame captive bred bearded dragon in your north American climate, if wild predators don't kill it , it'll freeze to death in the first frost.. I've been doing horse stuff for a good long time. I'm still excited about going to the barn like it's my first day. Yes, unfortunately, with certain irresponsible children and certain particularly sadistic parents, many pets do get thrown out. However, fortunately, I've grown more responsible over years, and my parents are too kind and caring to heartlessly throw out a cold-blooded creature that's not even native to where we live. If the now impossible thing happens and I grow "bored" with him (how is that even a thing?), the worst my parents will do is give him to good hands. I remember when I used to keep snails, then once I didn't do my chores. My mum took away the jar where I kept them and pretended to flush them down the toilet. I was so upset over that my mum couldn't hold back anymore and gave them back the next day.
There are risks to it's safety if there are other family pets (especially cats and yes even dogs can be a risky around bearded dragons , cats and lizards are a recipy for disaster especially, and if your rural / farm community is anything like those here, there at LOTS of cats and dogs about).