It is unfortunate that a driver with 20 yrs is trying to solicit a $9 per hour peak hire into doing something that violates the contract, violates company policy, and violates the law (I am assuming you do not have a D.O.T card.)
Under no circumstances is a helper allowed to deliver packages out of his/her personal vehicle. EVER. There are a some rules and policies that can be "bent" a little bit depending upon the situation, but this is NOT one of them.
You need to contact the management person who hired you and explain the situation to them. You want to work, but you dont want to break the law. Dont worry about getting your driver in trouble....if he is trying to coerce you into delivering packages out of your personal vehicle, he NEEDS to get in trouble.