My parents don't like sex, nudity, profanity, so Chinese dramas seem perfect for them. But they also don't like mystical anything (fortune telling, wuxia, etc.), so it's hard to get them to watch anything that isn't American. They adore, however, The Longest Day in Chang'an. I need something to show them afterward we get through it. I'm thinking When We Were Young could be worth a try, but any additional suggestions will be warmly appreciated. 

They are open to Korean dramas (my dad does like zombies, so he enjoyed Kingdom), but I'm studying Putonghua, so anything mainland Chinese would be preferable. They only speak English, so subtitles are a must.

If they're fine with medical drama, I can recommend Surgeons https://mydramalist.com/20536-surgeons . It share a lot of themes with Nirvana in Fire. 

Taiwanese daily dramas are for you!  They are all about family and promote family love and moral behavior in a gentle non-preachy way. Warning, they run long with many, many episodes. Try Inborn Pair. At 80 episodes it is a long journey but by the end I wished there there had been more.