You have two homes; therefore you would fast when you are in either place even if you don’t stay there ten full days.
However, when you are traveling from one home to the other home, if you leaver after Dhuhr time, then you keep your fast. If you leave before Dhuhur like 9 am and arrive before Dhuhr like 10 am then you keep your fast as well. Only if you leave before Dhuhr and arrive the other home after Dhuhr then you’d have to break your fast.
Having said that, there are two points here:
1- We are assuming that you are actually traveling from one home to the other home. If both of these homes are in one city, like New York or Chicago or London, then this is not considered traveling and you’d keep fasting in all scenarios.
2- If you are a frequent traveler then you keep your fast even if you are traveling. So if you are traveling twice every week, then you are a frequent traveler and you would keep your fast.
As for your husband he is able to keep his fast if he travels after Dhuhr. Now regarding the Isha prayer, he has to pray Qasr unless he is a frequent traveler (he travels to your parent’s house) several times a week. In that case he’d pray full.