Is it permissible to pay the zakat on hoarded gold on behalf of the father, without his knowledge, and is the son sinful for keeping his father's money without paying its zakat?

You are not obligated to pay Zakat on behalf of your father, and doing so would not fulfill his obligation. You should advise him and explain the religious ruling to him with gentleness and wisdom. There is no sin upon you for keeping his money, as this does not constitute direct or intentional assistance in his refusal to pay Zakat.