If the petitioner has been in their home country for about five months, this would generally be considered a "temporary stay" abroad. In technical terms, your parents are not living in their home country, but temporarily staying in their home country. In that case, their Green Card alone is sufficient to demonstrate that their domicile is in the United States. If additional proof is required, a valid state ID or driver's license should be more than enough.
However, if during the interview the applicant mistakenly states that the petitioner is currently living in their home country rather than in the U.S., it may raise concerns. In such a situation, even the documents listed above might not be sufficient. Domicile is not necessarily where you are currently living, but rather where your primary residence and intent to return permanently are located.
Regarding the second question: If a lawful permanent resident stays outside the U.S. for more than six months, they may still be allowed to return, but it could put their status at risk of being considered abandoned. However, this does not automatically affect their domicile claim until an immigration judge in US formally determines that they have lost their status.