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Tshirt and a nice cupcake shaped skirt, or tone it down with less poof. Keep with the knee high socks and some cute shoes. A tshirt usually makes it casual/daily lolita for me at least...

It's just nothing extravagant lolita, like no lace wristlets, no lace topped socks, no big head pieces or big bows, simple skirts, nothing with lots of lace or intricate stuff.

I would definatly not wear a petticoat for a while. Knee socks with skirts almost always are acceptable, since your just keeping your legs warm! simple headbands add a touch of lolita to an outfit. If you see a shirt or blouse at a "normal" store that you can wear for lolita, go ahead and get it, and start wearing it with jeans! At that point, its just a cute shirt you got at Forever 21, or where ever. As you start to incorporate more lolita-able items, it will get less and less strange to your parents. Just make sure that every day, when you look in the mirror, you can identify that little lolita element.

What style are you planning on wearing? EGL is very different from sweet which is very different from punk etc.... so the elements you need to focus on bringing into your wardrobe are going to be different. Remember as you start this process to get them used to the color pallet you'll be wearing as well. That was the shock for my parents - from bright colors to mostly black and white. xD

Cat: As much as I love helping you out, you need to learn how to do a little of your own research, too. This topic has been discussed a lot here and all you need to do is click "previous pages" to look for relevant topics.

http://www.gaiaonline.com/guilds/viewtopic.php?t=1254395&page=1
http://www.gaiaonline.com/guilds/viewtopic.php?t=14291589&page=1

These are from another Lolita Guild that actually isn't active right now. They still have some good posts up that you can look through.
http://www.gaiaonline.com/guilds/viewtopic.php?t=13290923

http://community.livejournal.com/daily_lolita/ is a good website to check out some casual outfits. Usually people post where their clothes are from, too.

I agree with Alpha Drug: Cute print t-shirts with cupcake shaped, full, skirts and kneehigh socks with flats usually make a cute outfit. Forever21 as mentioned before is also a good source, along with Charlotte Russe and H&M.

You could try to talk with your parents about this lovely new fashion you just discovered.

My suggestion usually includes grades and money. Like "hey mom, I got an A on my paper, can I wear a lolita outfit tomorrow?" "hey mom/dad, if I babysit and make money for myself, if I use money I earned myself, can I buy whatever I want?" <--that one worked for me, I got a job at Michael's at 16yo to support my hobby.

The other thing (that worked on my parents) was making comparisons to popular fashion: "hey mom/dad, what I'm wearing is modest and cute! I could wear booty shorts from Hollister/American Eagle; will you go buy me those slutty clothes instead, please?" Then again I was a snarky child.

Sorry, it was about 4am when I posted. I must have misread your post.

Still, Daily_lolita usually has some really cute casual lolita stuff that might help you out.