score:4
I do not think there is any connection in any Church document or scripture to the annunciation.
However, the sacraments have age requirements attached to them as a "rule of thumb." A certain level of maturity and knowledge is required to receive a sacrament. The Church could assign any age it deems necessary.
Even though the ages are 14 and 16 according to Canon Law, I think you would be hard pressed to find a Deacon or Priest in the developed world who would marry a couple this young.