Definitely +1 to Oh My Venus and here are some other recs:
Healer - I have not seen action- romance dramas dropped by your parents yet, so I thought I’d try. This one had me so hooked, I even got my mom to watch it and she actually finished it. Sad backstory though (I have not watched Kill me heal me, so I don’t know how they compare).
Some office romances:
Romance is a Bonus Book - This is a 10 to me, the office related issues actually interested me, but I don’t think it is a super exciting drama, it’s very mellow, so there is that to consider.
She would never know - Office romance but a bit more on the soap opera side.
Start up - Not exactly office romance since the get go. But it is business related and romance. Sad backstory unfortunately haha.
Dinner Mate - Also not exactly an office romance, but with plenty of office related moments, plus adult protagonists, not in their twenties.
Refresh Man - Not Korean, I know you said your parents are not into Chinese dramas, but this one is Taiwanese, so, I had to recommend it since it’s one of my fave office romances.