The standard deduction for a dependent is the GREATER of $1100 or your earned income + $350 . In your case you standard deduction is $5750. TurboTax automatically calculates that for you.
this assumes that the $5400 is your net self employment income, after deduction all your expenses.
You parents are half right, you do not need to worry about income tax, whether you are a dependent or not. But you will have to pay self employment tax (social security and medicare tax) on your net self employment income (about 14% net). Your self employment tax (SET)is the same whether you are a dependent or not. Your standard deduction does not help reduce SET.
With the tax law change, effective 2018, most students will get the same refund whether they claim themselves or not. The personal exemption has been eliminated and the standard deduction increased.
