Advice for securing an auto loan before I start my first full-time job?

Hi, I'm a recent college grad looking to buy a used car before I start work in 3 weeks. However, the first auto loan company I called said they couldn't work with me without proof of income from a current job, and the offer letter for the job I will be starting isn't sufficient.

I've worked internships and part-time jobs in the past. However, there are gaps in my employment and the pay from most wasn't high. My credit score is ~780. My parents gave me $5k for a downpayment, and I was looking at cars in the $13k to $16k range. I live 15 min away from my workplace, but there's no convenient public transport.

Thanks for reading and for any advice you have.