Is it permissible for the wife to reside in her parents' house after an agreement with her husband and his departure for work, despite there being no social, financial, or housing motive necessitating it?

There is no objection to the wife staying at her parents' house when her husband travels; in fact, it may often be the preferable and more suitable option, unless there is a detriment in her staying with them or a significant benefit in her remaining in her husband's home. The spouses should communicate and consider each other's circumstances, striving for what is in their best interest, both religiously and customarily.