The Form 1040 (U.S. tax return form) Instructions has a section called "Do You Have To File?". Below a certain income, you are not required to file a tax return and pay any tax. This amount of income at which you are required to file depends on several things, including your dependency status (you are a dependent of your parents), your marital status, and other factors. The instructions have charts that show what these numbers are. You would fall under Chart B.
Assuming that you are under age 65, unmarried, and not blind, you only have to file when you reach the following conditions:
Your unearned income was over $1,050.
Your earned income was over $6,300.
Your gross income was more than the larger of—
$1,050, or
Your earned income (up to $5,950) plus $350.
(Note: Income from YouTube would count as "earned income" for the purposes above.)
However, if you are producing your own videos and receiving revenue from them, you are technically self-employed. This means that the conditions from Chart C also apply, which state:
You must file a return if any of the five conditions below apply for 2015.
You had net earnings from self-employment of at least $400.
As a self-employed person, you can deduct business expenses (expenses that you incur in producing your product, which is this case is your videos). Once your revenue minus your expenses reach $400, you will need to file an income tax return.