I don't want to be morbid, but how ailing is the parent's health? Short-term or long-term? How much help are they truly in need of? Sometimes it's hard to tell from afar exactly what the situation is there, especially if it's dementia-related, or a new diagnosis. Are you going to show up and find that they need a full-time caregiver? Or is it more like "would be nice to have you drive us to the doctor a few times a year and mow the lawn" and a good excuse to move back to a location you like better? This should factor into your decision.
Your wife may be afraid that she will be forced into a caregiver role with your parents, especially if she moves without a job. Is it a guarantee that she could go Fed? How much do you need her income? How did she feel about you applying for this job? How would your relationship fare with the separation, if you go that route? All factors to consider.