Here is my collection of really bad Korean-English puns. Some original. Some from other places. Feel free to add your own in the comments.
What is Dracula's favorite drink?
What do you say if you don't have enough money at the pojang macha?
"오댕!" ("Oh dang!")
What do you bring to a party in the rain?
비 와요. 비. (BYOB)
When is the best time to catch a snowman?
After 눈 (Afternoon)
What country has four arms?
Why don't you care about a small cow?
It's so-so. 소소
What do you call a large piece of tissue?
In what country do you roll on the floor?
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Which fish goes to high school?
Godung-oh Hakkyo 고등어학교
What did the mother turkey say to the little turkey?
What did the lady say to the cute child with no ears?
Gui opda! 귀업다
What sound does a stale loaf of bread make?
What did the bell say in the bathroom?
What car does a cow drive?
Why couldn't the bean answer the phone?
콩비지 (Kong Busy)
How old is the average onion?
How do you send a tooth through the mail?
Me: Hey, Jian.
Jian: Hi, Daddy.
Me: How's your shoulder?
Me: I know it's 어깨, but how is it?
Let's buy a lion. 사자 사자
Heard about the accidental Korean cannibal?
He thought he was eating a salami sandwich.
What do dogs eat at Turkish restaurants?
I like soy milk, do you?
Where do you learn about tea?
From your tea차
Baechu were expecting cabbage.
Why did the librarian take a walk on the mountain?
He just wanted a sanchek (산책).
Why is a quilt always bad?
It's ee-vil 이불
Where can you find a tired pig?
Pig on hay, 피곤해
Mother cookie to her kid when it was time to go.