Is a wife obligated to obey her husband by staying away from his family to avoid problems with his overbearing mother, and by traveling with him to work abroad due to his difficult work circumstances, while still considering her parents' rights? What is the best way to show filial piety to her parents in this situation without falling short?

You must obey your husband and accompany him wherever he resides, as long as the place is safe and there is no harm to you. A husband may travel with his wife wherever he wishes, as long as she is safe and did not stipulate that he remain in her country. Obedience to the husband in this matter takes precedence over obedience to parents, because the husband's right is more obligatory. As Ibn Taymiyyah mentioned, the husband has more authority over his wife than her parents. Your staying away from your family does not prevent you from honoring them, as honoring encompasses all possible forms of kindness, such as financial assistance, calling, and gifting, and is not limited to residing with them or visiting them.