Both parties? I don't understand.
Your "tax home" is not the same thing as your permanent residence, or domicile. Your domicile is your permanent home. There is no single factor that determines domicile, it is a combination of all facts and circumstances. It includes things like, where do you live most of the time; where are your significant social relationships like friends or church; where are you registered to vote and drive; where are your significant professional relationships like doctor and dentist; and where you intend to return after any temporary assignments. There is no set time you must be away from your domicile before it changes, it's all about intention and action. Establishing a new domicile also requires taking active steps to abandon your prior domicile (like selling your home, canceling a lease. etc.). If your domicile is at your parents house, you could be away for months or even years, traveling on temporary assignments. As long as you consider your parent's house to be your permanent home, and you intend to return, and you never take steps to abandon that domicile, then you are probably still domiciled there.
Domicile is important for things like state income tax. You generally owe state income tax on all your world-wide income to the state where your permanent home or domicile is located. If you work or live temporarily in another state, you may also owe taxes to that state as a non-resident.
The concept of tax home is mainly used to determine if travel and transportation expenses are deductible for self-employed or independent contractors. (Tax home doesn't have much effect on the income tax of a W-2 employee). Your tax home is where you work most of the time, or where you earn most of your money. For example, I used to work with someone whose family lived in city A. He drove 90 miles to city B one-way for work. He stayed in a studio apartment, worked 10 hours monday-thursday, then drove home for a 3 day weekend with his family. City A was his domicile, but city B was his tax home. This meant that he could not deduct his travel expenses between A and B.
Tax home is discussed more in publication 463 on travel and transportation expenses.
If you are a travel nurse, there is no particular problem if you want to consider your parent's house your domicile. And it doesn't affect their taxes at all. But your tax home will generally be where you actually work, which may impact the deductibility of travel expenses. See publication 463 for more details and examples.
From publication 463 https://www.irs.gov/publications/p463
Tax Home
To determine whether you are traveling away from home, you must first determine the location of your tax home.
Generally, your tax home is your regular place of business or post of duty, regardless of where you maintain your family home. It includes the entire city or general area in which your business or work is located.
If you have more than one regular place of business, your tax home is your main place of business. See Main place of business or work, later.
If you don’t have a regular or a main place of business because of the nature of your work, then your tax home may be the place where you regularly live. See No main place of business or work, later.
If you don’t have a regular or main place of business or post of duty and there is no place where you regularly live, you are considered an itinerant (a transient) and your tax home is wherever you work. As an itinerant, you can’t claim a travel expense deduction because you are never considered to be traveling away from home.
