You seem to be confusing "modern" cars with "cars with massive, gas guzzling engines".
This is a norm for Americans. They have to have a car with a large engine because they think a small engine can't perform. It shocks people when my MINI can hang with larger vehicles.
If I had your budget, I'd be straight down to the Ford dealer and ordering myself a Fiesta. These are great cars, with sensible engines (even if you can't get the most sensible 1.25L version in the USA) and they look really modern. They have some great features, like the power heated windscreen (frost, snow, condensation is gone from the front windscreen at the push of a button), rear parking camera and also the option of a full leather interior.
If you were in Europe, I'd also be suggesting the Volkswagen Polo:
Heck, you're lucky enough to be able to afford the Audi A1:
All great vehicles that I had the luxury to drive while living in Germany. Some of the options that you have in Europe we don't have here, but for the most part they are nice cars.
Also, in the UK the insurance companies describe what you are planning to do as "fronting" and it can invalidate the insurance completely. That is to get your parents to sign for a policy as the "main" driver and then have you as a "named" or "second" driver - even though you will be driving the car more than them and they might not drive it at all. It is a very common form of fraud and they are cracking down on it. Not sure what the rules are in the USA. Plus, if you are not the policyholder then you won't earn any no claims bonus to use in future when you get your own policy.
This is allowed in the US. I'm not sure why the UK doesn't allow this but here in the USA, the parents can put the child on their policy and it usually save some money. That being said, the first accident that happens and the parents policy will go up and a possibility of being dropped by the insurance company.