I agree with those who suggest asking your parents for their credit card, if they want you to make their hotel arrangements - or offer to email information on nearby hotels with contact info to allow them to choose and reserve the room.
If they object to either offer, it would be pretty clear, from my POV, that they expect you to pay for the room, perhaps expecting this as your "hospitality" for their visit.
If you are okay with paying for the room in lieu of visiting them, I would make that clear - you have a set amount budgeted over the year for purposes of seeing them, do they prefer the hotel in your new area of residence, or that you visit them in their country this year - make it an either/or and let them decide, but make clear that "both" is not an option.
If you reserve a room on your credit card, their is no assurance that your parents will take the necessary action to have their credit card charged for the stay, and you could be stuck.