Are there any dramas where the parents of the guy are really supportive of the relationship and really love the girl as if they were their own child?

Something like the mom in My ID is Gangnam Beauty or the parents in Love O2O.

I'd love it if the parents have a bit larger role than in most dramas too.

They exist?

JK...Not completed but I would say:

What's Wrong With Secretary Kim. 

  • Bromance - ML' s whole family is extremely supportive; FL's cousin is also extremely supportive, but her parents are wary at first because the ML is the head of a triad family (which is now clean / legit) and they also generally (not related specifically to the ML) are concerned about the FL being in a relationship because they believe that the FL's life will be in danger if it's revealed that she is a girl (they've raised her to live as a boy) before her 26th birthday - they get over the initial concern relatively quickly, but the second concern remains a sticking point for a while; both families are quite involved in the storyline; the ML's family and the FL's cousin more than the FL's parents
  • In Time With You - both families support the leads becoming a couple well before they actually get there, IIRC 
  • Something About 1 Percent - the ML's grandfather likes the FL so much after she helps him once that he re-writes his will to try and force his grandson to marry her; I can't remember anything about the FL's family
  • Because This is My First Life - IIRC, the leads were expecting their parents to be opposed to them getting married (it's secretly a marriage of convenience), but both sets of parents are unexpectedly supportive 
  • The Beauty Inside - the first day the FL's mum meets the ML she tells him to marry the FL and the next time she sees the FL she tells her to marry the ML; the ML's mother and grandfather are against the couple at first but quickly make a hard turn and aggressively try to get the ML and FL together again after they fake break up (of course they're really dating for real in secret)

Also:

  • Itazura na kiss - and all its versions
  • Doctors
  • Love Now
  • Reply 1994
  • Legendary witch - ML is orphaned but his dead wife's dad who he now considers as his father totally likes the FL
  • Parting Left / goodbye to goodbye - mom would totally pick the girl than he son
  • Delightful girl chunhyang
  • Five enough - the mom loved yeontae for sangmin (oldest son) but sadly she followed the evil mom-in-law route for her second son taemin with GF Jinjoo
  • Ohisama (one of the kindest and supportive mother-in-law I've come across knew ahead who she wanted her son to marry...) 
  • I second 'Itazura na Kiss' all versions.

Angel Eyes - ML's mother likes FL very much.

The person I love - ML's family is very supportive of FL and is on her side rather than his.

Sud Sai Pan, the ML's family helps them a lot get then stay together. Her familyis rather neutral.

Kill me Heal me, her parents are adorbs. He's an orphan.