You need to have a social security number to have a job. Your parents probably already have this for you (it's required for them when paying taxes).
You do not need to show your new employer your social security card. You can show them your passport and that show proof of eligibilty to work in the US. When you start work, you need to complete a document called the I-9 that shows proof of eligibility to work in the US. On that paperwork it says you can show a passport OR social security card OR a bunch of other documents.
Your employer NEVER needs to see your social security card. Your problem is you are 16 and how do you make them see what they are requesting is wrong? Here's a copy of the I-9 paperwork.http://www.uscis.gov/files/form/i-9.pdf
One idea is for you & your parents to highlight the parts where it says a passport is good enough.
you can use the passport for your I-9 and you don't need anything else....
you do NOT have to provide your social security card to complete your W-4 although you do need to know your SS#......
that settles the legalities BUT
if this employer is requiring a SS Card (for whatever reason) then you will have to provide it,
there is no way for us to know why they have told you that you must have provide your SS Card....
if they have doubts about your number provide a check stub from your previous employer (if the number is on it) or a copy of your parents 1040 (white out everything except your name and SS# on the copy of the 1040)