Article: Often, parents will agree if you give them a little of what they want. Such an example would be: if you dye your hair and it turns out badly, you'll dye it back to its original color. Tell them "I'm fully prepared to dye it back to my original color if I don't like the color or it looks bad." If your parents are worried about you potentially messing up the dying process by doing it on your own or with a friend, this might be a good option.  Say "If you're worried about it being done badly, I could always get it done professionally instead. Then there's no worry about the quality of the outcome." The only downside to this is  that getting your hair dyed professionally will cost a good deal more. Offer that you both have to agree on the color you get. That allows you both to be in control. You could say "I'll try a more natural color close to my hair color first if you'd be more comfortable with that." Instead of dying your whole head, ask to put in highlights, lowlights, or streaks. Purple can still pack a punch from underneath your natural color. If you have long hair, then you can compromise with just dying ends. If it doesn't turn out well, or if you or your parents don't like it, then you can always cut it off. You can say something like "Maybe instead of dyeing the whole thing, I can just dye the ends. That way it's less different and it can always be cut off if it doesn't work out." If they are adamant against dying your real hair, suggest buying and coloring clip-in extensions so they can see how the color would look. It's not permanent and you can easily change the look if you or your parents don't like it. Another alternative is to use colored hair chalk, which washes out in the shower.
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Let them know you'll dye it back if it goes badly. Suggest getting it done professionally. Allow them to have some input on the color choice. Ask if you can partially dye your hair instead. Ask if you can get colored extensions instead.