I have a small question regarding Zakat: My mother has 113 grams of gold. She is a housewife, and my father has a personal loan. In this case, would Zakat be mandatory for us?
Because the amount of gold your mother owns exceeds the Nisab, then zakat has to be paid. If someone else (e.g. relatives) pays it on her behalf, there is nothing wrong with that.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: Is zakaah due on gold that is prepared for adornment, if a woman does not have any means of paying zakaah other than selling some of the gold?
The correct scholarly view and the most likely to be correct in my view is that zakaah must be paid on jewellery if it reaches the nisaab (minimum threshold), which is eighty-five grams. If it reaches this amount, zakaah must be paid on it. If she has other wealth and pays from that, there is nothing wrong with it. If her husband or one of her relatives pays it on her behalf, there is nothing wrong with that. If neither of these options is available to her, then she should sell some of it and pay zakaah with that money.
-Majmoo’ Fataawa Ibn ‘Uthaymeen, 18/138
Because your father is a debtor, he can accept zakat from others. As a result, you can pay your mother's zakat using the zakat your father earns.
Not paying zakat is highly punishable as Allaah (SWT) says in the Quran:
And those who hoard up gold and silver (Al‑Kanz: the money, the Zakaah of which has not been paid) and spend them not in the way of Allaah, announce unto them a painful torment.
On the Day when that (Al‑Kanz: money, gold and silver, the Zakaah of which has not been paid) will be heated in the fire of Hell and with it will be branded their foreheads, their flanks, and their backs, (and it will be said unto them:) ‘This is the treasure which you hoarded for yourselves. Now taste of what you used to hoard’”
To sum up, you are required to pay zakat of your mother's gold. If you don't have the money to pay the zakat, either you have to request help from your friends/relatives, or you have to sell a part of your mother's gold, or your father can accept zakat from other people and pay the zakat of your mother's gold.