I'm going through the process myself (nearly there!), having only one registered parent. I haven't seen anything about a new rule reducing the required stay for overseas nationals with two registered parents (I haven't been looking, either), though I'm sure a call to TECO will confirm whether or not it's true.
In any case, apply for your TARC as soon as you arrive, because the clock doesn't start ticking on your required stay (for your shenfenzheng) until it's issued.
And make sure all of your necessary documents are authenticated by the relevant TECO office. (So you have to get the TECO office that represents the area in which you were born to notarize your birth certificate.) Your parents will also need to submit their documents, so have those ready as well (you probably only need their shenfenzhengs and hukou certificate).
Taiwanese bureaucracy is really something, but you'll find that most of the people you'll have to deal with, regardless of actual competence and helpfulness, will be very friendly and patient as you work your way through the process...