Does a جاریہ (perpetual) charity reach a deceased parent by organizing a Quran memorization competition and placing it in the balance of their good deeds? And is it better to do it secretly or publicly? And is it permissible to fund its prizes from zakat money?

Every act of righteousness, the reward of which is gifted to the deceased, benefits them. Charity given on behalf of the deceased undoubtedly benefits them, so there is no harm in performing the actions you mentioned and gifting their reward to your father. It is preferable to conceal the good deed unless its public disclosure leads to a greater overriding benefit. It is not permissible to do this from zakat money, as zakat has specific designated recipients determined by Allah.