Hi Evie,
I would talk to your dad about your pet as well, though as LizardMom suggested, if your mother can take care of her "grandlizard" over the weekends, that might be the best solution. The fact that you are not certain if you are a fit caretaker implies that you have likely put a lot of thought into it already...
Here's me advice - Go through and make sure you have the funding for everything your beardie might need, including food and yearly veterinary visits. Make a beardie budget and talk through it with your parents, maybe post it on here to make sure you aren't forgetting anything. If you are a responsible kiddo - and it sounds like you are - making this list will likely reveal anything you aren't prepared to handle.
On that note, at 13 you're ~5 years from potentially going to college... Depending on where you go, lizards may not be allowed on campus. You should prepare for that eventually as well, since these guys can live longer than that, and/or anything similar like if your family is moving or if you have interest in working/studying abroad or joining the millitary, etc. Again, if your parents can care for the lizard for a couple years if need be, you're probably good.
My guess is that you would be a perfect beardie owner, though. Do keep us updated either way! And if you have any questions, let me know.