Sure you can do it on your own nickel. But this sort of behind-the-back stuff erodes trust. Is it really worth it? Why not just agree with them that the religious affiliation is a problem - and look to see if there are student groups at the school with a affiliation that they are more comfortable with. Maybe they’d be more willing for you to apply if the deal is that 1) should you be accepted and 2) you can afford to go, then in return 3) you will actively affiliate with a group that they are more comfortable with. Perhaps they’ll be willing to negotiate if you are.
By the way, just so you know, I wouldn’t pay for my kid to attend a school where I wasn’t comfortable with the religious affiliation or political orientation without being firmly convinced that it was the best option available and that my kid wasn’t going to be the odd person out at that school (and therefore under a lot of social pressure to conform). So I don’t think your parents are being unreasonable here. (But negotiation is healthier than ‘behind-the-back’ behavior.)