If your parents are footing the bill, they do have a say in your choices. On campus housing in most cases is more expensive than off campus housing, since you are required to have a meal plan which can get pretty pricey. Moving off campus in the LA/SB area can be expensive, but you can split the costs of rent/utilities/food etc… with several roommates. Have them check the COA of each campus comparing on-campus vs off campus housing. There are many ways to cut costs in your 2nd-4th year.
Here is a link showing the cost of attendance for UCLA:: Note the lower cost estimate for living off campus
https://www.admissions.ucla.edu/prospect/budget.htm
Here is a link for cost of attendance at UCSB:: Again note the lower cost estimate for living off campus
http://www.finaid.ucsb.edu/cost-of-attendance