Here is the full list of all 35 Spelunker Party! achievements.
Cleared 1-1.
Cleared all of World 1.
Cleared all of World 2.
Cleared all of World 3.
Cleared all of World 4.
Cleared all of World EX.
Cleared 10 stages with Spelunker.
Cleared 10 stages with Spelunkette.
Cleared 10 stages with Dark Spelunker.
Cleared 10 stages with Spelunkette's Sister.
Cleared with no errors for the first time.
Added 1 item to the Item Box.
Added 10 items to the Item Box.
Added 50 items to the Item Box.
Added 100 items to the Item Box.
Added all items to the Item Box.
Raised an item's level for the first time.
Raised an item to its max level.
Reached level 10.
Reached level 50.
Reached level 100.
Tried a quest for the first time.
Cleared 1 quest.
Cleared all rank 1 quests.
Cleared all rank 2 quests.
Cleared all rank 3 quests.
Cleared all rank 4 quests.
Cleared all rank 5 quests.
Played Explore Together and cleared for the first time.
Played and cleared with 100 people.
Got 1 NICE award.
Got 1 MVP vote.
Got 100 MVP votes.
Rescued someone for the first time.
Rescued others 100 times.