Postnatal Care Center (2020)

Both have themes that revolve around woman in a care center focused on treatment of pregnancy issues.

both Birthcare Centre and Avengers Social Club are female centered and "black" comedies, ie my type of story

Both Search: WWW and Birthcare Center are comedies and female centered dramas from TvN. Both have different story but I believe that you'll love Search: WWW if you liked Birtcare Center.

Different parents with different stories and parenting styles
Befriending each other and learning from said experiences
Both series are heartwarming and short

the comedy sense is kinda similar between Birthcare Center and Pegasus Market though, Birthcare Center is slightly less crazy xD

both Birthcare Center and Spring Turns to Spring are female centered, both have Uhm Ji Won and both are hilarious

Both Birthcare Centre and Oh My Baby are comedies from tvN about career female leads that are trying to have a baby.

Birthcare Centre is comedy drama about career female lead and she's trying to have a baby.
SKY Castle doesn't have much in common. It's more focused on education than mothers/parents and their childs. But it's also female centred plot.

both deal with serious postpartum issues and the burdensome expectation the society has s on mothers

Birthcare Center has a more mature approach considering it's showing the life of mothers with newborn babies. Both dramas have great sismance vibes and showing females supporting each other. They have great comedy but also give meaningful messages. Actor Yoon Park played in both dramas.