if my father transfers shares purchased from HIS Zerodha account to MINE (Son), who will have to pay the STCG/LTCG whenever I (son) sell those shares from MY account?
The transfer of shares is normal and not in form of gift. both of us use Zerodha account.
A parent can give any amount of shares, money to his children, they are considered as gift, and the capital gains or losses incurred from the gift are the child’s. So if you made profit after receiving them as gift, you pay tax.
As @Suyash.K has correctly answered, since the shares have been transferred by your father to you without any adequate consideration, it will be considered as gift given. Now, when you sell these shares, gain on transfer shall be taxable in your hands under income from capital gains.