Is it Islamically permissible to leave one's job and return to Yemen to care for elderly parents—given the security unrest in Yemen, the mother's previous refusal, and the father's lack of an explicit request—and will Allah compensate for the financial hardship?

If your return to your country is for the sake of honoring your parents, then your action is religiously sound and you are a doer of good. Allah does not let the reward of those who do good go to waste, and it is hoped that Allah will replace what you left with something better, as He جل وعلا says: "And whoever fears Allah - He will make for him a way out And will provide for him from where he does not expect." And if your situation becomes difficult in your country, and you wish to travel, and you hope that it will be better for you and your family, and you do not fear any harm to your parents by your absence from them, then there is no blame upon you.