Should a husband pay the agreed-upon price of gold to his wife, or should he take out a loan to bring his mother, who can afford her travel expenses?

If the gold is part of the Mahr (dowry), which is an obligatory payment, then your husband is not permitted to prioritize his mother's ticket over it, because repaying a debt is obligatory. However, if it is merely a promise, then it is commendable for him to fulfill it, and to prioritize what is closer to the greater good between buying the gold and his mother's ticket. A righteous wife should assist her husband in honoring his parents, and it is not good for her to prevent him from bringing his mother.