You might be surprised- just because someone is very religious does not mean that they won't accept and support ->-bleeped-<-. There are links everywhere, and a lot of people go through this or know someone that has gone through it. I was so surprised when my [very Christian and very Midwestern] mom's side of the family accepted me with open arms- I had been thinking I would be somewhat ridiculed when I came out to them... but not at all.
I second the suggestions to ask for therapy, but it is all in the delivery of the request that matters right now because you don't want to upset things at home. Maybe instead of focusing on the fact that what you feel inside is that you are female, focus on explaining to them the growing anxiety it causes you in life and the need to talk with someone experienced about it.
Maybe even say something to them like, "I need to know if what I feel inside of me is real, because I can sense that it is beginning to feel like a burden and I want to nip it in the bud now before it becomes a problem"... So basically avoiding the parts that they are not ready to believe yet and instead talking about a growing discomfort you have with yourself. Avoid being so decidedly trans, and they will likely let you have your therapy in hopes that it will "fix" you. Even if your goal is to not be "fixed" in the way that they would like (for you to remain cis male), a therapist is the first and strongest step towards proper treatment. Depending on where you live, therapy IS the first door you have to walk through if you do decide to transition. So, you might as well at least try to ring the doorbell now

In the case with loving parents (which you obviously have) they almost always do come to accept you- and it doesn't even have to be hard. The therapist is your ticket in, because I agree that hearing it from a professional is likely going to help them accept it sooner.
In the meantime try to allow them to see the discomfort it causes you, but don't at all become a burden on them. If they can easily see you are still trying to succeed and are failing or frustrated, they will become concerned. However conversely if they see you becoming a ball of negativity that complains about everything, they probably won't like it too much. Either way as loving parents they will eventually have to give in and let you have your therapy. You have more say in it than you might imagine.
Don't worry, stay strong, and thoughtfully communicate your discomfort to your parents. It may take a little effort on your part, but over time they will get it and you will be able to get the ball rolling. I can tell that you are a sweet individual, and because of that I am very confident you're going to get through this just fine