Just last week I remembered that Seorak National Park often has trail closures during the muddy season. I looked into it and the highest peak, Dae-chung Bong, will be closed from April 4 until May 13. Other trails deep in the park will also be closed.
I believe that short trails near Seorak-dong will remain open so you can still get to Ulsan Bowi or Gum Gang Gul (cave). Both are short but rewarding hikes with fantastic views.
Odaesan may also have trails closed. Last year, I went to So Gum-gang with my full hiking gear. I knew I didn't have time to cross the park but I planned to go a few hours in, then back out again. I had lots of water and food, my hiking stick and other bits of gear, only to go thirty minutes in and finding my way blocked. The waterfall at the turn-around point was beautiful but I had hoped to go much further.
Oh, a comment on Korean place names. I thought I was hiking at 'Salt River' (So-gum Gang) and no one understood. Finally, I learned that the area is called 'Little Gum-gang' (So Gum-gang). Gum-gang Mountain, in North Korea, is so beautiful (so they tell me) that many places in Gangwon Province try to cash in on the name.