Serious KDramas to Watch with Parents

So as the title says I'm looking for Kdramas to watch with my Dad specifically as Im trying to get him into watching korean shows more... My Mom and I always watch Kdramas together but we watch the fluffy romantic stuff that we all love and adore however my Dad isn't the biggest fan of all that fluff xD.. However my family and I always have movie nights together as well as watch series together (but english series and not asian) and my taste in films are the same as my Dad and I'm very well aware of his preferences but I stumbled onto the Drama "My Mister" and this was the very first time I'm seeing such a drama coming from SK and I instantly knew this is the kinda drama my Dad would watch with us and enjoy and so we started a wonderful new routine of watching episodes together with some coffee and snacks in hand til sunrise and even my siblings joined in and loved it!

My Dad loved that show so I was hoping if anyone would suggest a series I could watch with him again? I got my Dad into korean films recently too (especially the thrillers and more serious ones such as 1987 when the day comes and The Chaser) but I would love to know a new kdrama that we can start watching together..
to give you a general idea of my Dads taste. Movies and series:

My Mister, Peaky Blinders, Forest Gump, Green Book, Leon the professional, The Blind Side, The Scent of a Woman, Gladiator, The Godfather, 1987:When the day comes etc.

Its okay if there's romance but just not the overwhelming kind but more of a mature one (Mature in the sense of the characters personalities and not in the other sense of course so don't misunderstand xD)

Please help me continue my lovely routine with my parents and show them the beautiful things Asian films/dramas have to offer!

@Forsteye Wow You hit the nail right on the head! From reading the synopsis of some of these, they are exactly the kind of dramas I've been looking for so Thank you SO SO MUCH, it definitely helped! ^-^

I remember I watched Bad Guys with my Dad a few years back and he did enjoy it so I cant wait to dive into your suggestions and get started.

• +1 Mr. Sunshine
• Itaewon Class - Includes a good father / son relationship, although the father dies. 
• Hello Monster
• Secret
• Mad Dog - Includes a good father figure relationship. Mostly serious but sometimes a bit cheesy.
• Misty - There are a few scenes that might be a little uncomfortable with your dad depending on your tolerance for watching intimate scenes in his presence. There's no nudity (a few scenes with shirtless men) but there are some moments and flashbacks that are supposed to be sexy.
• The K2 - The FL and romance aren't all that mature, but the ML is an ex-mercenary out for revenge and most of the story is serious. The secondary female lead / villain is mature. Watch this scene ahead of time to see if you would be comfortable watching it with your dad.

 elena:

Chicago Typewriter, Hymn of Death & Mr Sunshine.  I can't vouch for the quality because I've only seen part of Chicago Typewriter

I personally wouldn't recommend Hymn of Death. The ML is married to someone other than the FL, he has a bad relationship with his father, and he and the FL commit suicide together, so it might not be great for watching with family. I also didn't enjoy the leads' relationship and felt that some parts were quite boring.