Absolutely apply! My feeling is that most colleges are absurdly overpriced and nobody should just pay full price if they don’t have to, regardless of their financial situation. Most colleges have different "buckets’ of money for need-based and merit-based aid anyway, so you are not taking money away from a more deserving student if you qualify. As for outside competitive scholarships, unless they have need-based criteria I think they expect students from all economic backgrounds to apply. I see nothing unethical about trying to save your parents money!
My alma mater state U. has a “legacy tuition” program that awards scholarships effectively giving in-state tuition to out of state children of alumni, based on their academic merit. The university is doing this to try to entice students to come back to the state and hopefully stay there. If my D chooses to attend that college, I would certainly have her apply for that scholarship, even though it is one of the lowest OOS tuition state universities out there.
I admire your for stopping to think and consider this. Back in dark ages when I went to college, my parents would not allow me to apply for federal financial aid because “it was no one’s business how much money we have”. I also could not apply for local scholarships in my hometown for the same reason. They were self-employed, had minimal savings, and I likely would have qualified. Rather, I was told as a HS freshman that I needed to do well in high school, try to be Valedictorian and get a scholarship to the state flagship university. I did that which paid tuition and I worked summers to make money to pay for room and board (which was much, much less then, of course). I thought that was pretty much what everyone did. I was not prepared to go to college and hear girls in the dormitory saying, “Oh, financial aid checks came in today, let’s drive to the mall and go shopping!” I guess there will always be people who abuse the financial aid system, which is unfortunate. As the folks ahead of me have said, if it is a merit scholarship and you can earn it, go for it!