oh man, that's always rough.
explain to your parents that these dolls aren't 'just for playing', or even just for display, but valuable collector's items and also valuable art pieces. and they really are beautiful works of sculpture.
the way i explained mine to my parents (the first one, anyway - they fell in love with them afterwards

) is that i was also looking to improve my photography, and this is something i am motivated to photograph. gotta have a subject to photograph if you're working to improve your photographs, right?
also mention that it's a social hobby, and a social avenue, as well as an artistic one. you not only get a photography subject, but you get to join a vast community of people to share tips with and talk to and meet up with. and mention that it's a creative hobby in more than one respect: a lot of people learn how to sew and make props, and if you learn how to do it in 1/6 scale, you can do it in 1 to 1 lifesize scale too.

all else fails, gently explain to your parents thus: "i know you may not see why this is important to me, and why i like this so much, but this is my own money i'm spending, remember? even if you think it is a mistake, sometimes people need to make mistakes on their own so they can grow. i really am interested in this, though, and would like your okay to buy this thing i really want, because i don't want to upset you in any way. i know you see it as a waste of money, and i don't want to anger you so that you ever throw her away or anything. worst case scenario, i've researched and i can probably resell her for about the same investment if i ever get bored of my new doll. please understand that even if it IS a mistake, this might be something i need to do on my own to learn about myself and grow. i am not going to pressure you to give me even more money - i am happy saving up for more stuff. i've patiently saved up for this for quite awhile, because it is important to me, and a thing i really like and have looked forward to owning for awhile. and i would love your understanding and support in this, because i love you both and this is important to me."
parents generally cave if you remind them that you love them, and that they are important to you, and that's why you're seeking their help on something.

(because, i hasten to add, they know it's true. it's a good thing to love your parents, and to have them love you.) but basically, remind them that it's *your mistake to make*, and you understand the consequences of it. and that to continue to have it be 'your mistake', you aren't going to start demanding money from them - you're going to save up for anything else.
many times parents get a little vexingly controlling with money/what stuff you buy because they're just not used to you being so independent. i mean, i am Rather Old, and even now my dad has admitted to me that it's hard for him to think of me as substantially older than my wee newborn niece (his grandchild). of course it would be silly for a toddler to buy a bjd... ....but you're not a toddler. and having to confront how old you are (and thus how old THEY are) is a heavy thing for a lot of parents. it's something kids rarely acknowledge as something that's scary or vexing for parents, because from our point of view as children it just seems like they're smothering us a little and not treating us appropriately. of course, that's also true, but i think sometimes gently acknowledging their fears helps a lot. "i know if i was little, this would be a bit of a silly purchase. but keep in mind american girl dolls with all their stuff cost about the same, right? and this is something i can enjoy far more for many years. i want this to be my introduction into a bit more of a grown-up hobby, since children don't collect these dolls. so this might be one of my first real adult purchases of things. i know you think it's a bit silly, but i've been saving up, and this is something i really do want."
i would also tease out their fears a little more. if they're afraid of you getting ripped off, which is understandable - "my kid wants to spend HOW much money on a weird little Chinese company i ain't heard of before?" - maybe try finding the same item through mintoncard or denver doll emporium, which are American companies that may be able to soothe that particular fear.
you might want to ask them point-blank what they think you should spend your money on instead. if there really is some need they're trying to direct you to, hear them out. and again, tell them, "this is my own money that i've saved up carefully. i'm not going to pressure you for more money, or use your money for this hobby, because it's going to be my hobby, not yours, after all. so please don't feel worried about financial stuff. i really do have the money for this, because i've wanted it for a long time." if there is a genuine need you're neglecting, deal with that first, of course. but i doubt you're not taking an animal to the vet to preserve dolly money or skipping a car repair bill to preserve dolly money or something equally silly there, of course. the point of this is to make your parent's fears and concerns feel validated and cared for. listening to them on this point gets them to get it off their chests so to speak. i actually bet what is happening is *they* are projecting their financial concerns on *you*, because there's a lot of financial unease around these days. you may want to offer point-blank to help out if your family is struggling with enough grocery money or something. your parents will probably turn you down, but they'll respect you for wanting to contribute like an adult. and if they don't, well, no doll for a little bit more, but eating is more important after all (and if they're struggling to feed you, that would explain why they're balking so hard at the luxury of a doll, of course).
all else fails, i'd put that doll in your bookmarks and start smaller. junkyspot has some really cute plastic, not resin, little bjds for like 30-40 dollars - little rabbits, mice, cats. consider getting one of these, and letting your parents see you painting the faceup, making the wig for it, sewing clothes for it, etc. - letting your parents see that it's a creative hobby and one you enjoy. once you get the door propped open that way, they'll be more willing to open it wider.
btw, the little hujoo abs plastic ones are hella cute. you can see mine, a little mouse called Bianca, in this rather clumsy shoot of my dolly shelf at the moment

so if you like bjds you might consider one anyway.
you would have to do the faceup yourself though since i believe junkyspot doesn't offer that service anymore (though you could shoot them an email and ask just in case!)