I am in a somewhat similar situation. I am not out to my very religious family. My father is an ordained pastor, actually.
I am an atheist, but I grew up with a Christian education, and these may or may not be useful to you in your explanation.
-the only two anti-transsexual scriptures and their reubuttals:
Deut 23:1
"No one whose testicles are crushed or whose penis is cut off shall be admitted to the assembly of the Lord."
- This Scripture is sometimes used to justify transphobia. However, from a literary perspective, it is fulfilled by the death of Christ on the cross in the New Testament, just as all the covering laws, code of the clean and unclean, and the sacrificial rituals of the OT were fulfilled.
Deut 22:5
"A woman shall not wear a man's apparel, nor shall a man put on a woman's garment; for whoever does such things is abhorrent to the Lord your God."
- This needs no rebuttal if your parents accept that you ARE a woman, not a man behaving "like" a woman. But, I will include a rebuttal if you are not so successful with them immediately. My argument is that cultural gender codes for dress vary between region, culture, and time period, and obeying this scripture is relative. Perhaps siting the prevalence of gender variance among cultures and time periods would be helpful.
Many contemporary Christians do not affirm the Old Testament. If this is the case, you can point out that no NT scripture condemns transgender or transsexual identities. In fact, two NT scriptures contradict Deut 23:1
Matthew 19:12
12For there are eunuchs who have been so from birth, and there are eunuchs who have been made eunuchs by others, and there are eunuchs who have made themselves eunuchs for the sake of the kingdom of heaven. Let anyone accept this who can."
Isaiah 56:4-5
4For thus says the Lord: To the eunuchs who keep my sabbaths, who choose the things that please me and hold fast my covenant, 5I will give, in my house and within my walls, a monument and a name better than sons and daughters; I will give them an everlasting name that shall not be cut off.
Perhaps the most important distinction you could make would be to explain in clear terms that gender identity/expression and sexual orientation are distinct categories and are not the same. A common misconception, especially among religiously motivated bigots, is that transsexuality is an extreme form of homosexuality. If you can explain this distinction, you may get far.
Even in the case that your parents do not accept your gender identity, they can be civil and loving. They are commanded not to judge you, and believe their god will forgive you of all your sins and love you unconditionally.
Best of luck.