DD went to school in SD from the east coast.
Her campus had ZipCars and the student account rate was very reasonable. Back then I think it was $8/hr. She would book a few hours with a friend, they’d do their shopping and split the fee, hassle free.
Check if that’s an option for you.
As someone who had a car in college, I could give you many reasons not to bring it if you plan on living in a dorm. To name a few . . .
If you don’t have a parking space lined up somewhere before you go, it may be tough. I was able to purchase a permit at an off campus office building lot, but it wasn’t cheap and took about 30 minutes to walk to. If you can’t line up a set space/lot, you’ll likely end up having to take your chances on the street where most end up with multiple expensive parking tickets and bumper scrapes from people pulling in and out in front or behind them.
Also, having it out in the elements (assuming you don’t have covered parking), door dings, becoming everyone’s chauffeur (gets old), and the fact that you’ll find you rarely actually use it (not enough time with all you’ll have going on) while paying what will likely be a high rate of insurance while it just sits somewhere. And what do you do with it during breaks? At the end of the year, it may not look like a show car anymore.
I would at least leave it home for the first semester/quarter until you can figure out what the parking issues are at and around your college and come up with a viable plan. If you decide to move off campus into an apartment complex after the first year, you may have covered/garage parking included and it makes it MUCH easier to have a car.
But since you sound dead set on taking it and your parents are dead set against you driving, ship it. By the end of the year after you’ve been away and navigating life across the country successfully on your own, I’d guess they may be more likely to let you have your adventure driving it back.
Good luck!
@1Dreamer I don’t think bad weather will ever be an issue in San Diego!
But to the OP…what will you do when your school year ends? Drive back home to NY?
Honestly…my Giess is your parents are saying NO to taking the car and you are hoping to gain some support for doing so.
Like I said…I have NOT heard a compelling reason why you must take a car to college 3000 miles from where your family lives.
@thumper1 , not snow or infrequent rain, but the baking sun does a number on the paint and interior. Also, if he’s talking about UC San Diego or Scripps by beach, combine that with the moist salt air. I lived on the beach for a while after college and within an hour after washing my car, it would be covered with a sticky sand dust film if it was parked outside. Even my brass lamps at home became pitted from the corrosive salt air. He says it’s like a show car, so I’m guessing he’s going to be a little obsessive about it and wanting to keep it that way.
But yes, I agree with you that it would be smarter NOT to take the car. 
I am also curious what school in San Diego??? Just asking because like others have pointed out, parking at UCSD is a pain. and may not even be available if you are a freshman. Parking is also expensive. Uber is available, the train is nearby, and you can definitely get by w/o a car. Most freshmen do not have cars there. Just something to consider before you decide to take yours- wherever you are going, check into parking on campus.
If you drive it across the country that’s a lot of mileage. If you treat your car like a show car then some maintenance will have to happen when you get across the country.
First of all let me start with the cons:
-Damn near 2800 miles (assuming no sight-seeing detours) means you might need an oil change pretty fast after you get there. If you use full synthetic that’s 65-100ish$ right there.
-If it’s a manual transmission, that’s 2800 miles of wear you put on your clutch.
-Lets assume you get about 25 MPG. 2800/25 is 112 gallons of gas. If it’s a show car, you better be putting premium in that thing to keep it running right. If we assume $3/a gallon that’s about $336 in gas.
-Parking costs can be high on campuses.
Pros:
-To really enjoy the San Diego area, it’s nice to have a car. If you’ve already got one then that makes it especially easy.
-If you love driving your car, that’s quite a bonding experience.
I’m not going to try to convince you to leave your car. I love my car, I would never leave it. That being said, I would strongly recommend you re-consider shipping. Pick a good shipping company, maybe buy some kind of soft cover, and ship it across country. There’s a lot of potential for issues if you drive it across country. Also costs you might not be thinking off, and I would strongly recommend against doing it by yourself. There’s lots of beautiful roads around San Diego to enjoy and the weather is good for all year round driving 
When Kiddo had a car in California, in order to get a neighborhood parking pass we were required to change the registration to California. We also could not use our regular insurance agent because they could not write insurance for a car that would be kept out-of-state. We had to go with an online insurance company.
Kiddo’s friend got tickets for parking his car with Texas plates in various places around town because he had been in the state for more than 30 days. The State didn’t care that either one of them were in the State because they were college students.
As part of that registration the car will have to be able to pass the smog test…
Why not fly to school, then wait a few months to assess whether you really need one there?
My kids don’t even want a car. They uber everywhere.
If you will be a typical resident student at UCSD, SDSU, or USD, better to leave the car at home the first term. It will be just one more thing to worry about while you are adjusting to college. When you get there, you can assess parking availability and cost, as well as other issues like changing your driver’s license, car registration, and car insurance to California if you start driving in California. If it is actually advantageous to bring your car, then you can drive it or have it shipped for a later term.
Also, with only 4,000 miles on it, there are special emissions-related rules to meet when bringing in a car from out-of-state. https://www.dmv.ca.gov/portal/wcm/connect/4a6b05aa-1e88-4e31-ac02-453e3a71c7c7/ffvr29.pdf?MOD=AJPERES
With newer cars, there may also be tax consequences for CA car registration. Also lots of D’s classmates were issued expensive tickets if they didn’t register their out of state cars with CA DMV and get CA insurance.
So far, have read NO good reasons for you bringing a car to school but many great reasons for leaving it home.
Another issue you may encounter…so plan ahead. There are MANY hotels out there who will not allow an under 21 year old to reserve a room…or check on for one (and all require a photo ID when you check in).
You may not find a hotel where you (and a friend) can spend the two nights you will need to spend on the road.
Also, how will your friend get back? Will you pay for the airline ticket for,the friend to get back east?