Here is the full list of all 13 YANKAI'S PEAK. achievements.
Complete the first bridge.
Complete all levels in Part I.
Complete the second bridge.
Complete all levels in Part II and earlier.
Complete the third bridge.
Complete all levels in Part III and earlier.
Complete the fourth bridge.
Complete all levels in Part IV and earlier.
Complete the fifth bridge.
Complete all levels in Part V and earlier.
Complete the sixth bridge.
Complete all levels in Part VI and earlier.
Complete YANKAI'S PEAK.