I went through something similar.
I only had my mom growing up & when I was 16 I got a job & one of the first things I wanted since I was young was a snake. I begged/pleaded with her but she was deathly afraid of snakes. I promised she would never be forced to be around it & I would make sure it was only out in my room. She said NO! So I stopped pushing the fact.
I started going to the libary & local pet store and gathered information on the snake I wanted. I came to her after a few months & showed her all the info I had regarding it & she still said no! After that I gave up on it. A few months later for my Bday I got a nice surprise & baby burmese with everything that was needed to care for it.
Not saying it will work that way for you but it's worth a try to show them that you are willing to devote all the time to gather the info it may push them to say yes. One word of advice is not to push too hard, if they say no it's not the end of the world, one day you will be on your own then you don't have to ask them for permission to get one. It will all come in due time.
Reply With Quote