When many people hear you have a degree from a prestigious school, they automatically assume you're smart and talented and right about whatever you're talking about, even if it's unrelated to your degree. This is a general effect that goes far beyond individual fields like science or law. Just being aware of your school will cause many people to unconsciously defer to your opinions and automatically treat you like their social superior.
This effect is so pronounced, some people try to avoid saying where they went to school, instead saying things like 'I went to school in the Bay Area' or 'I went to school in Boston'. They find the automatic obseqious treatment they get awkward, so they only mention their school when they need to use it for something they want.
In practice, this general 'we automatically respect your opinions and assume you're right' effect makes getting all kinds of opportunities easier. Jobs, promotions, meetings with and mentorship from powerful people, etc. It can also help with petty bureaucrats, law enforcement, medical doctors - pretty much everywhere.
It doesn't surprise me that programmers say a prestigious degree doesn't matter - this is America, after all, and automatically treating someone as your better seems profoundly undemocratic. But saying you don't care about prestigious degrees is kind of like saying 'advertising doesn't work on me'. People who actually don't care about and react positively to prestigious degrees are very rare.
I'll add that I didn't know this effect existed when I was going to school - I just wanted to be a history professor! I only noticed it later, when I realized the game board of life was oddly slanted to favor me. But I bet your parents are very well aware of this, and that's why they want you to stay in your school.