Yours reacted exactly the same as mine did over 10 years ago, even down to some of their "rationalizations" (if you can call it that). Please don't hold this against yourself at all. And I really mean at all - not even your 'giving in' as you called it.
Your parents are having a massive, massive case of the Dunning Kruger effect. The answer is right in front of them and they just can't handle it, because it makes them feel like they have no power or understanding over you, their "creation" and charge.
Give them time. I found absolutely no amount of talking helpful with mine. Reading my father's journals, it was appalling what blasphemous misconceptions he had of me. Both of mine eventually came around, in their own way, to the extent that they feel able. That might be easier today where there's 10,000x more online coverage and resources of gender topics than back then.
They need time to process this; but moreso, I found in my case it was extremely helpful for them to see how happy and successful I was after transition, especially compared to what a train-wreck I was beforehand.