I have touched upon the topic of self esteem and dealing with society's reactions, yet I believe learning to deal with parents reactions can be just as daunting. For a young Lolita, the only things getting in your way may be money concerns and parents....
I get mixed emotions from my parents no matter what outfit I am wearing. Sometimes they hate my fashion, sometimes they love it. Although this can be the same with any outfit, and any fashion that anyone wears.
Whether you are trying to work up the courage to show off your first coordination, or you just need the courage to be able to get past the fear of what your parents may think, there are many different things you can do to help.
Research- This is one thing, I regret to inform you, I did not do. Before you purchase your first coordination, research as much as you possibly can, ask for advice online from other Lolita's and learn exactly what the fashion is as a whole. As you research you can show your mum or your dad some of the things you find. Find pictures of 'actual' people, and definitely not anime characters. Although some people get into Lolita Fashion through anime, the two have nothing to do with one another. Showing your parents pictures of anime characters will get them to believe that the fashion is cosplay, or something real people don't wear.
As you go through this process of research, show your parent or guardian that this is something you like. Point out that dress you see online, tell them you think it's cute. Point out that girl on Tumblr and say how you like what she wears. Just show them as much of the fashion as you can, and get them warmed up to the idea that this is something that you like.
Don't lie- After hinting at the fashion for a while and telling them how cute you think it is, you may want to embrace the first step of buying or eventually wearing your garments. I can't stress how important this point is, don't lie to your parents. If they are confused and don't understand something, do your best to explain it even if they don't understand or believe you at first. Don't say Lolita is cosplay just to get them to buy you something, as they will forever believe it. Don't sneak behind their backs and buy items they don't want you buying, or getting your friends to buy things for you. Even if your parents don't understand Lolita, being deceitful or lying to them is just going to make them ban it from you all together, or hate the fashion even more.
Budget- Some things parents hate about Lolita Fashion is the price. If brand is your thing, a parent may not understand that or want to buy these things for you. No way is any parent going to buy, or even see a reasonable reason for you buying that $800 dress. Start out with off brand for your first coordinate if you can, and show them this is something you love. Wear your off brand coordinate as much as you can and work your way up to the more expensive things gradually.
If you are old enough to get a job, then I highly recommend you do so. If you show your parents this is something you will work hard towards to obtain yourself, you are showing them you are responsible. If you are not old enough to get a job there are other ways you can still get the things you want. Collect birthday and Christmas money, do odd jobs around the house, ask to baby sit a younger sibling or relative etc. Even if you can't earn much, work hard at it to achieve what you want.
Be gradual- If you aren't a very feminine girl then of course your parents won't want you to buy that Sweet Lolita dress, or see any point in you wearing something at all feminine, elegant or cute. If you are always wearing band t-shits and skinny jeans for example, your parent may just groan and see this as a phase you will grow out of, and a total waste of time and money to buy these clothes.
Although it may seem like a pain, the best idea is to gradually introduce more feminine or Lolita like outfits into your wardrobe. And even if you do own Lolita already, start wearing dressier clothes when going out with mum just to prove that you can be girly when you want to. Show your parent of guardian this is something you love, and you will do whatever it takes to get it.
Unfortunately though no amount of advice I can give will entirely change a parents mind. Some people get lucky and have parents that love Lolita Fashion for various reasons, others have parents who don't care what they do, and some have parents who hate and despise Lolita Fashion to no end. Unfortunately we can not change a parents opinion sometimes, and while we live under their roof we have no choice but to abide by their rules.