TARCs are issued to individuals (often overseas Taiwanese, spouses, and recently naturalized Taiwanese) who don't have a household registration 户籍 and thus do not have a National ID number. For overseas Taiwanese, you need to obtain an overseas Taiwanese passport from the Taiwanese Economic/Culture Office (TECO) in the UK, enter Taiwan on that passport, and then apply for the TARC. You will need a bunch of documents for the TARC including your authenticated and translated birth certificate, background clearance (FBI check for the U.S., not sure what it is for the UK), parents' marriage certificate, and parents' household (户籍) registration. You will have to process all these first at the UK TECO consulate before coming to Taiwan. Also, you have to get a medical certificate (physical) in Taiwan.
I'm not sure if you can apply for the TARC if your name is on your parents' household registration. The next step would be to contact the nearest TECO. Good luck.