Is it permissible for me to pay zakat on the money I borrowed from my father for the purpose of marriage, on his behalf, to alleviate his financial burden?

The zakat on debt may be obligatory on both the creditor and the debtor, or it may be specific to one of them.

Zakat becomes due on the creditor annually, upon the completion of a year (hawl), if the debtor is capable and acknowledges the debt. If the debtor is incapable or denies the debt, then no zakat is due on the creditor until he collects it, whereupon he pays zakat for each past year, according to the majority of scholars.

As for the debtor, the majority of jurists hold that debt negates zakat on intrinsic wealth if it encompasses or reduces it below the nisab. If he possesses other wealth not subject to zakat (such as real estate for personal use), he can designate it against the debt to exempt his zakatable assets and then pay their zakat.

If zakat becomes obligatory on both you and your father, you may voluntarily pay your father's zakat on his behalf each year, provided it is with his permission, as intention is a prerequisite for zakat to be valid.