Cradle Catholic here, too. My mom comes from an uber-Catholic family and my dad converted so he could marry her. While both espouse traditional Catholic views on social issues, my mom seems to be a bit more welcoming and my dad, eh, let's just say he's tolerant. And I love them both! As for me, I still consider myself a practicing Catholic and that my faith is important to me. I say all this just to give you a bit of a background and to say that I truly get the struggle you are going through.
So, advice? Not to sugar coat it, but it's not going to be easy to come out to your parents--but I guess that depends largely on your parents. Perhaps you may be surprised. What I did was sort of easy my way into it. I had an inkling that my mom would be a bit more welcoming so I came out to her first. She had lots of questions and there was a lot of crying but I know my mom and I know how to hedge my bets with her. I also made it a huge point to make her feel like I understood what she was thinking and feeling. I asked her not to tell my dad about this until she and I could have more time figuring out what this meant for one another. That took about 2-3 months of small conversations that really seemed to work. Then I spoke to my dad, and, yeah. That went as expected. Ironically, he was more mad at my mom for not telling him--or at least that's how it seemed like he was projecting his frustration. Again, it took being totally open and honest and lots of crying over 6ish months for us to come to a point of 'normalcy'. So my process wasn't like pulling a Band-Aid--short, painful. In the end, it seems to have worked for me. At least when I return home, they aren't piling the firewood and getting ready to burn me at the stake. Now, extended uber-Catholic family--well, that's a different story.
And the priest part. If you Catholic faith is important to you, it's worth the struggle to talk to him. He will tell you the things you expect to hear from a Catholic priest, but through the process you may come to have a better sense of self and understand how you can reconcile your faith and who you are. And, again, this isn't a quick process--I'm still struggling with this but the struggle is worth it. Furthermore, given my background, I know the Catholic church doesn't accept 'transgender' so I can't force it like "YOU MUST ACCEPT ME AS IS OR FUCK YOU!"--as much as I have felt like that sometimes. After talking to my priest for years about it, he's said to me that he still rejects the fact that I am a 'woman', but he at least understands what I am going through and it's helped him be able to "separate the sin from the sinner" and minister to the soul. I doubt he will change his mind in regards to me, but maybe the next transgender parishioner he talks to, he will be better positioned to help them. At least that's my hope.
I hope this can help a bit.