If $15,000 really is the max your parents will pay annually for college, I would suggest you start with University of Alabama, University of New Mexico, and Arizona. It’s very possible you will get sufficient merit aid at those to make the schools affordable.
I would suggest you read through this thread. Like you, the family had a firm price limit and was looking for merit aid as they didn’t qualify for need based aid. This student wanted engineering, IIRC. But you can check some of the recommendations to see if CS is offered.