I was in your same boat, my mom wasn't for it at either, I wanted a either a green cheek or a sun conure. If I were you I would wait. First because you don't know what you want for sure, I've been reading your post and not that long ago you were going to get a cockatiel, then conure, now a parakeet. I would give it until you know for sure what you want. Then go from there, after making up your mind 100% start buying the the thing you'll need for it. Once you have everything for it, tell them your getting it. If they are aware of you buying stuff, they'll see how much you want a bird, and unless they stop you from buying the thing you'll need for it, odds are when you say alright I'm getting the bird they'll let you.
P.S. At the time I was putting everything together to get the sun conure I was on winter break, and didn't have a job. When I got a job a month later and the spring semester started, I thought it was best to wait as I wouldn't of had much time to spend with it. You should think ahead months, years down the road because if you plans or life change your poor bird will feel the side effects. Hope all go's well.