Is it necessary for the future in-laws of the two places to exchange greetings on holidays? Is it appropriate for the man's parents to say hello and the other party doesn't respond?

It's up to you to call your in-laws later. It doesn't matter if you say hello here and the other party doesn't respond. Maybe the other party doesn't have this habit, or the other party's economic conditions are not very good. As long as you feel reasonable, keep doing it. As for what the other party thinks, it's their business.