I got a really good financial offer from Rowan State University and almost nothing from Rider but I want to attend Rider. My parents think I should call Rider and tell them what Rowan is offering me. Does that work if I have no other circumstances (no one lost their job or anything, I just want more money). Thank you.
Your parents are right. (Of course 
) But don’t make a cold call. First set up an appointment to meet with a financial aid officer at Rider. If you are in New Jersey (and it sounds like you are), an in-person appointment would be best (and you can bring your parents, too, if you wish). If you can’t get to campus, a phone appointment is fine.
Ask for a specific amount of money, not just for “More” (like Oliver Twist!). Try to come up with a figure that would work for your family but that isn’t so sky-high that the Rider folks will just laugh.
Even if there are no circumstances that clearly warrant this request (lost job, illness, etc.), it would be helpful if you can arrive armed with figures that suggest that you have greater need than your Expected Family Contribution suggests. (e.g., invoices for mortgage payments, recent home repairs, medical expenses, etc.) BUT … if your family’s biggest bills are for your Christmas in the Caribbean or the new Tesla in the garage, then leave the bills at home!
If your long-term, post-college plans sound altruistic but low-paying (e.g., you want to join the Peace Corps or teach special-needs preschoolers), you can also mention that you are wary of taking on debt because you’re aiming for a field that won’t be lucrative.
Definitely present your aid award from Rowan at this meeting and explain that Rider remains your top choice, but that you will have to attend Rowan if Rider cannot provide some additional aid. But, in doing this, beware of acting ENTITLED. Instead, you need to seem grateful for your Rider acceptance and for whatever pittance they’ve already offered. If you come off sounding as if you DESERVE more, it will work against you. Admission officials often have some wiggle-room when deciding if they can give out extra funds, and the students who appear humble usually fare best in the appeals process.